Algebraic Combinatorics

AlgebraicCombinatorics.FPS.EssentiallyFinite f = EssentiallyFinite f

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Definition 1.79

A (possibly infinite) family $\left(\mathbf{a}_{i}\right)_{i\in I}$ of FPSs is said to be summable (or entrywise essentially finite) if

\[ \text{for each }n\in \mathbb {N}\text{, all but finitely many }i\in I\text{ satisfy }\left[x^{n}\right]\mathbf{a}_{i}=0. \]

In this case, the sum $\sum _{i\in I}\mathbf{a}_{i}$ is defined to be the FPS with

\[ \left[x^{n}\right]\left(\sum _{i\in I}\mathbf{a}_{i}\right) =\underbrace{\sum _{i\in I}\left[x^{n}\right]\mathbf{a}_{i}}_{\substack {\text{an essentially}\\ \text{finite sum}}} \ \ \ \ \ \ \ \ \ \ \text{for all }n\in \mathbb {N}\text{.} \]

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def AlgebraicCombinatorics.FPS.SummableFPS {R : Type u_1} [CommRing R] {ι : Type u_2} (f : ι → PowerSeries R) :

AlgebraicCombinatorics.FPS.SummableFPS f = ∀ (n : ℕ), {i : ι | (PowerSeries.coeff n) (f i) ≠ 0}.Finite

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noncomputable def AlgebraicCombinatorics.FPS.summableFPSSum {R : Type u_1} [CommRing R] {ι : Type u_2} (f : ι → PowerSeries R) (_hf : SummableFPS f) :

PowerSeries R

AlgebraicCombinatorics.FPS.summableFPSSum f _hf = PowerSeries.mk fun (n : ℕ) => ∑ᶠ (i : ι), (PowerSeries.coeff n) (f i)

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Proposition 1.80

Let $\left(\mathbf{a}_{i}\right)_{i\in I}$ be a summable family of FPSs. Then, any subfamily of $\left(\mathbf{a}_{i}\right)_{i\in I}$ is summable as well.

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theorem AlgebraicCombinatorics.FPS.summableFPS_subfamily {R : Type u_1} [CommRing R] {ι : Type u_2} {f : ι → PowerSeries R} (J : Set ι) (hf : SummableFPS f) :

SummableFPS fun (i : ↑J) => f ↑i

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Proposition 1.81

Sums of summable families of FPSs satisfy the usual rules for sums (such as the breaking-apart rule $\sum _{i\in S}a_{s}=\sum _{i\in X}a_{s}+\sum _{i\in Y}a_{s}$ when a set $S$ is the union of two disjoint sets $X$ and $Y$). See [ 19s , Proposition 7.2.11 ] for details. Again, the only caveat is about interchange of summation signs: The equality

\[ \sum _{i\in I}\ \ \sum _{j\in J}\mathbf{a}_{i,j}=\sum _{j\in J}\ \ \sum _{i\in I}\mathbf{a}_{i,j} \]

holds when the family $\left(\mathbf{a}_{i,j}\right)_{\left(i,j\right) \in I\times J}$ is summable (i.e., when for each $n\in \mathbb {N}$, all but finitely many pairs $\left(i,j\right) \in I\times J$ satisfy $\left[x^{n}\right]\mathbf{a}_{i,j}=0$); it does not generally hold if we merely assume that the sums $\sum _{i\in I}\ \ \sum _{j\in J}\mathbf{a}_{i,j}$ and $\sum _{j\in J}\ \ \sum _{i\in I}\mathbf{a}_{i,j}$ are summable.

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theorem AlgebraicCombinatorics.FPS.summableFPS_fubini {R : Type u_1} [CommRing R] {ι : Type u_2} {κ : Type u_3} {f : ι × κ → PowerSeries R} (hf : SummableFPS f) :

(∀ (i : ι), SummableFPS fun (j : κ) => f (i, j)) ∧ ∀ (j : κ), SummableFPS fun (i : ι) => f (i, j)

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Definition 1.82

Let $x$ denote the FPS $\left(0,1,0,0,0,\ldots \right)$. In other words, let $x$ denote the FPS with $\left[x^{1}\right]x=1$ and $\left[x^{i}\right]x=0$ for all $i\neq 1$.

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theorem AlgebraicCombinatorics.FPS.X_coeff_one {R : Type u_1} [CommRing R] :

(PowerSeries.coeff 1) PowerSeries.X = 1

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theorem AlgebraicCombinatorics.FPS.X_coeff_ne_one {R : Type u_1} [CommRing R] (i : ℕ) (hi : i ≠ 1) :

(PowerSeries.coeff i) PowerSeries.X = 0

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Lemma 1.83

Let $\mathbf{a}=\left(a_{0},a_{1},a_{2},\ldots \right)$ be an FPS. Then, $x\cdot \mathbf{a}=\left(0,a_{0},a_{1},a_{2},\ldots \right)$.

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theorem AlgebraicCombinatorics.FPS.X_mul_shift {R : Type u_1} [CommRing R] (f : PowerSeries R) (n : ℕ) :

(PowerSeries.coeff (n + 1)) (PowerSeries.X * f) = (PowerSeries.coeff n) f

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theorem AlgebraicCombinatorics.FPS.X_mul_coeff_zero {R : Type u_1} [CommRing R] (f : PowerSeries R) :

(PowerSeries.coeff 0) (PowerSeries.X * f) = 0

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theorem AlgebraicCombinatorics.FPS.coeff_X_mul {R : Type u_1} [CommRing R] (f : PowerSeries R) (n : ℕ) :

(PowerSeries.coeff n) (PowerSeries.X * f) = if n = 0 then 0 else (PowerSeries.coeff (n - 1)) f

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Proposition 1.84

We have

\[ x^{k}=\left(\underbrace{0,0,\ldots ,0}_{k\text{ times}},1,0,0,0,\ldots \right) \ \ \ \ \ \ \ \ \ \ \text{for each }k\in \mathbb {N}. \]

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theorem AlgebraicCombinatorics.FPS.X_pow_coeff {R : Type u_1} [CommRing R] (k n : ℕ) :

(PowerSeries.coeff n) (PowerSeries.X ^ k) = if n = k then 1 else 0

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Corollary 1.85

Any FPS $\left(a_{0},a_{1},a_{2},\ldots \right) \in K\left[\left[x\right]\right]$ satisfies

\[ \left(a_{0},a_{1},a_{2},\ldots \right) = a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots = \sum _{n\in \mathbb {N}}a_{n}x^{n}. \]

In particular, the right hand side here is well-defined, i.e., the family $\left(a_{n}x^{n}\right)_{n\in \mathbb {N}}$ is summable.

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theorem AlgebraicCombinatorics.FPS.fps_eq_tsum_coeff {R : Type u_1} [CommRing R] (f : PowerSeries R) :

f = PowerSeries.mk fun (n : ℕ) => (PowerSeries.coeff n) f

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theorem AlgebraicCombinatorics.FPS.summableFPS_monomial_family {R : Type u_1} [CommRing R] (a : ℕ → R) :

SummableFPS fun (n : ℕ) => PowerSeries.C (a n) * PowerSeries.X ^ n

The Chu–Vandermonde identity

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Proposition 1.104

Let $a,b\in \mathbb {N}$, and let $n\in \mathbb {N}$. Then,

\[ \binom {a+b}{n}=\sum _{k=0}^{n}\binom {a}{k}\binom {b}{n-k}. \]

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theorem AlgebraicCombinatorics.FPS.vandermonde_nat (a b n : ℕ) :

(a + b).choose n = ∑ ij ∈ Finset.antidiagonal n, a.choose ij.1 * b.choose ij.2

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Theorem 1.106 Vandermonde convolution identity, aka Chu–Vandermonde identity

Let $a,b\in \mathbb {C}$, and let $n\in \mathbb {N}$. Then,

\[ \binom {a+b}{n}=\sum _{k=0}^{n}\binom {a}{k}\binom {b}{n-k}. \]

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theorem AlgebraicCombinatorics.FPS.chuVandermonde {S : Type u_2} [CommRing S] [BinomialRing S] (a b : S) (n : ℕ) :

Ring.choose (a + b) n = ∑ ij ∈ Finset.antidiagonal n, Ring.choose a ij.1 * Ring.choose b ij.2

Dividing FPSs

Inverses in commutative rings

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Definition 1.107

Let $L$ be a commutative ring. Let $a\in L$. Then:

(a) An inverse (or multiplicative inverse) of $a$ means an element $b\in L$ such that $ab=ba=1$.

(b) We say that $a$ is invertible in $L$ (or a unit of $L$) if $a$ has an inverse.

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def AlgebraicCombinatorics.FPS.IsInverse {L : Type u_1} [CommRing L] (a b : L) :

AlgebraicCombinatorics.FPS.IsInverse a b = (a * b = 1)

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theorem AlgebraicCombinatorics.inverse_unique {L : Type u_2} [CommRing L] {a b c : L} (hb : a * b = 1) (hc : a * c = 1) :

b = c

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Theorem 1.108

Let $L$ be a commutative ring. Let $a\in L$. Then, there is at most one inverse of $a$.

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theorem AlgebraicCombinatorics.inverse_unique {L : Type u_2} [CommRing L] {a b c : L} (hb : a * b = 1) (hc : a * c = 1) :

b = c

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Definition 1.109

Let $L$ be a commutative ring. Let $a\in L$. Assume that $a$ is invertible. Then:

(a) The inverse of $a$ is called $a^{-1}$.

(b) For any $b\in L$, the product $b\cdot a^{-1}$ is called $\frac{b}{a}$ (or $b/a$).

(c) For any negative integer $n$, we define $a^{n}$ to be $\left( a^{-1}\right)^{-n}$. Thus, the $n$-th power $a^{n}$ is defined for each $n\in \mathbb {Z}$.

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def AlgebraicCombinatorics.divByUnit {L : Type u_2} [CommRing L] (b : L) (a : Lˣ) :

AlgebraicCombinatorics.divByUnit b a = b * ↑a⁻¹

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Proposition 1.110

Let $L$ be a commutative ring. Then:

(a) Any invertible element $a\in L$ satisfies $a^{-1}=1/a$.

(b) For any invertible elements $a,b\in L$, the element $ab$ is invertible as well, and satisfies $\left(ab\right)^{-1}=b^{-1}a^{-1}=a^{-1}b^{-1}$.

(c) If $a\in L$ is invertible, and if $n\in \mathbb {Z}$ is arbitrary, then $a^{-n}=\left(a^{-1}\right)^{n}=\left(a^{n}\right)^{-1}$.

(d) Laws of exponents hold for negative exponents as well: If $a$ and $b$ are invertible elements of $L$, then

\begin{align*} a^{n+m} & =a^{n}a^{m}\ \ \ \ \ \ \ \ \ \ \text{for all }n,m\in \mathbb {Z};\\ \left( ab\right) ^{n} & =a^{n}b^{n}\ \ \ \ \ \ \ \ \ \ \text{for all }n\in \mathbb {Z};\\ \left( a^{n}\right) ^{m} & =a^{nm}\ \ \ \ \ \ \ \ \ \ \text{for all }n,m\in \mathbb {Z}. \end{align*}

(e) Laws of fractions hold: If $a$ and $c$ are two invertible elements of $L$, and if $b$ and $d$ are any two elements of $L$, then

\[ \frac{b}{a}+\frac{d}{c}=\frac{bc+ad}{ac}\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \frac{b}{a}\cdot \frac{d}{c}=\frac{bd}{ac}. \]

(f) Division undoes multiplication: If $a,b,c$ are three elements of $L$ with $a$ being invertible, then the equality $\frac{c}{a}=b$ is equivalent to $c=ab$.

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theorem AlgebraicCombinatorics.fracs1_inv_eq_one_mul_inv {L : Type u_2} [CommRing L] (u : Lˣ) :

u⁻¹ = 1 * u⁻¹

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theorem AlgebraicCombinatorics.fracs1_isUnit_mul {L : Type u_2} [CommRing L] {a b : L} (ha : IsUnit a) (hb : IsUnit b) :

IsUnit (a * b)

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theorem AlgebraicCombinatorics.fracs1_mul_inv_rev {L : Type u_2} [CommRing L] (u v : Lˣ) :

(u * v)⁻¹ = v⁻¹ * u⁻¹

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theorem AlgebraicCombinatorics.fracs1_mul_inv_comm {L : Type u_2} [CommRing L] (u v : Lˣ) :

(u * v)⁻¹ = u⁻¹ * v⁻¹

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theorem AlgebraicCombinatorics.fracs1_zpow_neg_eq_inv_zpow {L : Type u_2} [CommRing L] (u : Lˣ) (n : ℤ) :

u ^ (-n) = u⁻¹ ^ n

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theorem AlgebraicCombinatorics.fracs1_zpow_add {L : Type u_2} [CommRing L] (u : Lˣ) (n m : ℤ) :

u ^ (n + m) = u ^ n * u ^ m

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theorem AlgebraicCombinatorics.fracs1_mul_zpow {L : Type u_2} [CommRing L] (u v : Lˣ) (n : ℤ) :

(u * v) ^ n = u ^ n * v ^ n

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theorem AlgebraicCombinatorics.fracs1_zpow_mul {L : Type u_2} [CommRing L] (u : Lˣ) (n m : ℤ) :

(u ^ n) ^ m = u ^ (n * m)

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theorem AlgebraicCombinatorics.fracs1_div_add_div {L : Type u_2} [CommRing L] (a c : Lˣ) (b d : L) :

divByUnit b a + divByUnit d c = divByUnit (b * ↑c + ↑a * d) (a * c)

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theorem AlgebraicCombinatorics.fracs1_div_mul_div {L : Type u_2} [CommRing L] (a c : Lˣ) (b d : L) :

divByUnit b a * divByUnit d c = divByUnit (b * d) (a * c)

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theorem AlgebraicCombinatorics.fracs1_div_eq_iff {L : Type u_2} [CommRing L] (a : Lˣ) (b c : L) :

divByUnit c a = b ↔ c = ↑a * b

Inverses in $K\left[ \left[ x\right] \right] $

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Proposition 1.111

Let $a\in K[[x]]$. Then, the FPS $a$ is invertible in $K[[x]]$ if and only if its constant term $[x^{0}]a$ is invertible in $K$.

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theorem AlgebraicCombinatorics.fps_invertible_iff_constantCoeff {K : Type u_1} [CommRing K] (a : PowerSeries K) :

IsUnit a ↔ IsUnit (PowerSeries.constantCoeff a)

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Corollary 1.112

Assume that $K$ is a field. Let $a\in K[[x]]$. Then, the FPS $a$ is invertible in $K[[x]]$ if and only if $[x^{0}]a\neq 0$.

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theorem AlgebraicCombinatorics.fps_invertible_iff_constantCoeff_ne_zero {F : Type u_2} [Field F] (a : PowerSeries F) :

IsUnit a ↔ PowerSeries.constantCoeff a ≠ 0

Newton's binomial formula

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Proposition 1.118

The FPS $1+x\in K[[x]]$ is invertible, and its inverse is

\[ \left(1+x\right)^{-1}=\sum _{n\in \mathbb {N}}\left(-1\right)^{n}x^{n}. \]

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theorem AlgebraicCombinatorics.fps_onePlusX_isUnit {K : Type u_1} [CommRing K] :

IsUnit (1 + PowerSeries.X)

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theorem AlgebraicCombinatorics.fps_onePlusX_inv {F : Type u_2} [Field F] :

(1 + PowerSeries.X)⁻¹ = PowerSeries.mk fun (n : ℕ) => (-1) ^ n

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Theorem 1.119

For each $n\in \mathbb {Z}$, we have

\[ \left(1+x\right)^{n}=\sum _{k\in \mathbb {N}}\binom {n}{k}x^{k}. \]

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theorem AlgebraicCombinatorics.fps_newtonBinomial_nat {R : Type u_2} [CommRing R] (n : ℕ) :

(1 + PowerSeries.X) ^ n = PowerSeries.mk fun (k : ℕ) => if k ≤ n then ↑(n.choose k) else 0

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theorem AlgebraicCombinatorics.fps_newtonBinomial_neg {F : Type u_2} [Field F] [BinomialRing F] (n : ℕ) :

(1 + PowerSeries.X)⁻¹ ^ n = PowerSeries.mk fun (k : ℕ) => Ring.choose (-↑↑n) k

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Theorem 1.120

Let $n\in \mathbb {C}$ and $k\in \mathbb {Z}$. Then,

\[ \binom {-n}{k}=\left(-1\right)^{k}\binom {k+n-1}{k}. \]

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theorem AlgebraicCombinatorics.binomUpperNegation {R : Type u_2} [CommRing R] [BinomialRing R] [NatPowAssoc R] (r : R) (k : ℕ) :

Ring.choose (-r) k = (-1) ^ k * Ring.choose (r + ↑k - 1) k

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theorem AlgebraicCombinatorics.binomUpperNegation_int (n : ℤ) (k : ℕ) :

Ring.choose (-n) k = (-1) ^ k * Ring.choose (↑k + n - 1) k

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Proposition 1.121

For each $n\in \mathbb {N}$, we have

\[ \left(1+x\right)^{-n}=\sum _{k\in \mathbb {N}}\left(-1\right)^{k}\binom {n+k-1}{k}x^{k}. \]

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theorem AlgebraicCombinatorics.fps_onePlusX_pow_neg {F : Type u_2} [Field F] [BinomialRing F] (n : ℕ) :

(1 + PowerSeries.X)⁻¹ ^ n = PowerSeries.mk fun (k : ℕ) => (-1) ^ k * ↑(Ring.choose (↑n + ↑k - 1) k)

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Corollary 1.122

For each $n\in \mathbb {N}$, we have

\[ \left(1+x\right)^{-n}=\sum _{k\in \mathbb {N}}\binom {-n}{k}x^{k}. \]

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theorem AlgebraicCombinatorics.fps_onePlusX_pow_neg' {F : Type u_2} [Field F] [BinomialRing F] (n : ℕ) :

(1 + PowerSeries.X)⁻¹ ^ n = PowerSeries.mk fun (k : ℕ) => Ring.choose (-↑↑n) k

Dividing by $x$

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Definition 1.123

Let $a=\left(a_{0},a_{1},a_{2},\ldots \right)$ be an FPS whose constant term $a_{0}$ is $0$. Then, $\frac{a}{x}$ is defined to be the FPS $\left(a_{1},a_{2},a_{3},\ldots \right)$.

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def AlgebraicCombinatorics.PowerSeries.divByX {K : Type u_1} [CommRing K] (a : PowerSeries K) :

PowerSeries.constantCoeff a = 0 → PowerSeries K

AlgebraicCombinatorics.PowerSeries.divByX a x✝ = PowerSeries.mk fun (n : ℕ) => (PowerSeries.coeff (n + 1)) a

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Proposition 1.125

Let $a,b\in K[[x]]$ be two FPSs. Then, $a=xb$ if and only if $\left[x^{0}\right]a=0$ and $b=\frac{a}{x}$.

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theorem AlgebraicCombinatorics.fps_eq_X_mul_iff {K : Type u_1} [CommRing K] (a b : PowerSeries K) :

a = PowerSeries.X * b ↔ PowerSeries.constantCoeff a = 0 ∧ ∃ (h : PowerSeries.constantCoeff a = 0), b = PowerSeries.divByX a h

A few lemmas

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Lemma 1.128

Let $a\in K[[x]]$ be an FPS with $\left[x^{0}\right]a=0$. Then, there exists an $h\in K[[x]]$ such that $a=xh$.

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theorem AlgebraicCombinatorics.fps_exists_X_mul_of_constantCoeff_zero {K : Type u_1} [CommRing K] (a : PowerSeries K) (ha : PowerSeries.constantCoeff a = 0) :

∃ (h : PowerSeries K), a = PowerSeries.X * h

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Lemma 1.129

Let $k\in \mathbb {N}$. Let $a\in K[[x]]$ be any FPS. Then, the first $k$ coefficients of the FPS $x^{k}a$ are $0$.

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theorem AlgebraicCombinatorics.fps_coeff_X_pow_mul_eq_zero {K : Type u_1} [CommRing K] (k : ℕ) (a : PowerSeries K) (m : ℕ) (hm : m < k) :

(PowerSeries.coeff m) (PowerSeries.X ^ k * a) = 0

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Lemma 1.130

Let $k\in \mathbb {N}$. Let $f\in K[[x]]$ be any FPS. Then, the first $k$ coefficients of $f$ are $0$ if and only if $f$ is a multiple of $x^{k}$.

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theorem AlgebraicCombinatorics.fps_first_k_coeffs_zero_iff_X_pow_dvd {K : Type u_1} [CommRing K] (k : ℕ) (f : PowerSeries K) :

(∀ m < k, (PowerSeries.coeff m) f = 0) ↔ PowerSeries.X ^ k ∣ f

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Lemma 1.131

Let $a,f,g\in K[[x]]$ be three FPSs. Let $n\in \mathbb {N}$. Assume that $[x^{m}]f=[x^{m}]g$ for each $m\in \{ 0,1,\ldots ,n\} $. Then, $[x^{m}](af)=[x^{m}](ag)$ for each $m\in \{ 0,1,\ldots ,n\} $.

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theorem AlgebraicCombinatorics.fps_coeff_mul_eq_of_coeff_eq {K : Type u_1} [CommRing K] (a f g : PowerSeries K) (n : ℕ) (h : ∀ m ≤ n, (PowerSeries.coeff m) f = (PowerSeries.coeff m) g) (m : ℕ) :

m ≤ n → (PowerSeries.coeff m) (a * f) = (PowerSeries.coeff m) (a * g)

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Lemma 1.132

Let $u,v\in K[[x]]$ be two FPSs such that $v$ is a multiple of $u$. Let $n\in \mathbb {N}$. Assume $[x^{m}]u=0$ for each $m\in \{ 0,1,\ldots ,n\} $. Then $[x^{m}]v=0$ for each $m\in \{ 0,1,\ldots ,n\} $.

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theorem AlgebraicCombinatorics.fps_coeff_zero_of_multiple {K : Type u_1} [CommRing K] (u v : PowerSeries K) (n : ℕ) (hdvd : u ∣ v) (hu : ∀ m ≤ n, (PowerSeries.coeff m) u = 0) (m : ℕ) :

m ≤ n → (PowerSeries.coeff m) v = 0

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Lemma 1.346

If $a\overset {x^n}{\equiv } b$ and $c\overset {x^n}{\equiv } d$, then $ac\overset {x^n}{\equiv } bd$. In other words, $x^n$-equivalence is preserved under multiplication.

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theorem AlgebraicCombinatorics.fps_coeff_mul_eq_of_both_coeff_eq {K : Type u_1} [CommRing K] (a b c d : PowerSeries K) (n : ℕ) (hab : ∀ m ≤ n, (PowerSeries.coeff m) a = (PowerSeries.coeff m) b) (hcd : ∀ m ≤ n, (PowerSeries.coeff m) c = (PowerSeries.coeff m) d) (m : ℕ) :

m ≤ n → (PowerSeries.coeff m) (a * c) = (PowerSeries.coeff m) (b * d)

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theorem AlgebraicCombinatorics.FPS.xnEquiv_mul_of_coeff_eq {K : Type u_1} [CommRing K] {n : ℕ} {a b c d : PowerSeries K} (hab : ∀ m ≤ n, (PowerSeries.coeff m) a = (PowerSeries.coeff m) b) (hcd : ∀ m ≤ n, (PowerSeries.coeff m) c = (PowerSeries.coeff m) d) (m : ℕ) :

m ≤ n → (PowerSeries.coeff m) (a * c) = (PowerSeries.coeff m) (b * d)

Polynomials

Definition

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Definition 1.134

(a) An FPS $a\in K\left[ \left[ x\right] \right] $ is said to be a polynomial if all but finitely many $n\in \mathbb {N}$ satisfy $\left[ x^{n}\right] a=0$ (that is, if all but finitely many coefficients of $a$ are $0$).

(b) We let $K\left[ x\right] $ be the set of all polynomials $a\in K\left[ \left[ x\right] \right] $. This set $K\left[ x\right] $ is a subring of $K\left[ \left[ x\right] \right] $ (according to Theorem 1.148 below), and is called the univariate polynomial ring over $K$.

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def FPS.IsPolynomial {K : Type u_1} [CommRing K] (f : PowerSeries K) :

FPS.IsPolynomial f = {n : ℕ | (PowerSeries.coeff n) f ≠ 0}.Finite

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def FPS.polynomialSubalgebra {K : Type u_1} [CommRing K] :

Subalgebra K (PowerSeries K)

FPS.polynomialSubalgebra = { carrier := {f : PowerSeries K | FPS.IsPolynomial f}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, algebraMap_mem' := ⋯ }

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Theorem 1.148

The set $K\left[ x\right] $ is a subring of $K\left[ \left[ x\right] \right] $ (that is, it is closed under addition, subtraction and multiplication, and contains the zero $\underline{0}$ and the unity $\underline{1}$) and is a $K$-submodule of $K\left[ \left[ x\right] \right] $ (that is, it is closed under addition and scaling by elements of $K$).

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def FPS.polynomialSubalgebra {K : Type u_1} [CommRing K] :

Subalgebra K (PowerSeries K)

FPS.polynomialSubalgebra = { carrier := {f : PowerSeries K | FPS.IsPolynomial f}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, algebraMap_mem' := ⋯ }

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def FPS.polynomialSubring {K : Type u_1} [CommRing K] :

Subring (PowerSeries K)

FPS.polynomialSubring = FPS.polynomialSubalgebra.toSubring

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def FPS.polynomialSubmodule {K : Type u_1} [CommRing K] :

Submodule K (PowerSeries K)

FPS.polynomialSubmodule = Subalgebra.toSubmodule FPS.polynomialSubalgebra

Reminders on rings and $K$-algebras

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Definition 1.151

The notion of a ring (also known as a noncommutative ring) is defined in the exact same way as we defined the notion of a commutative ring in Definition 1.42, except that the “Commutativity of multiplication” axiom is removed.

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theorem FPS.ring_add_assoc {R : Type u_2} [Ring R] (a b c : R) :

a + (b + c) = a + b + c

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theorem FPS.ring_add_comm {R : Type u_2} [Ring R] (a b : R) :

a + b = b + a

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theorem FPS.ring_add_zero {R : Type u_2} [Ring R] (a : R) :

a + 0 = a

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theorem FPS.ring_zero_add {R : Type u_2} [Ring R] (a : R) :

0 + a = a

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theorem FPS.ring_add_neg {R : Type u_2} [Ring R] (a : R) :

a + -a = 0

docs source

theorem FPS.ring_mul_assoc {R : Type u_2} [Ring R] (a b c : R) :

a * (b * c) = a * b * c

docs source

theorem FPS.ring_left_distrib {R : Type u_2} [Ring R] (a b c : R) :

a * (b + c) = a * b + a * c

docs source

theorem FPS.ring_right_distrib {R : Type u_2} [Ring R] (a b c : R) :

(a + b) * c = a * c + b * c

docs source

theorem FPS.ring_mul_one {R : Type u_2} [Ring R] (a : R) :

a * 1 = a

docs source

theorem FPS.ring_one_mul {R : Type u_2} [Ring R] (a : R) :

1 * a = a

docs source

theorem FPS.ring_mul_zero {R : Type u_2} [Ring R] (a : R) :

a * 0 = 0

docs source

theorem FPS.ring_zero_mul {R : Type u_2} [Ring R] (a : R) :

0 * a = 0

blueprint

Definition 1.152

A $K$-algebra is a set $A$ equipped with four maps

\begin{align*} \oplus & :A\times A\rightarrow A,\\ \ominus & :A\times A\rightarrow A,\\ \odot & :A\times A\rightarrow A,\\ \rightharpoonup & :K\times A\rightarrow A \end{align*}

and two elements $\overrightarrow {0}\in A$ and $\overrightarrow {1}\in A$ satisfying the following properties:

The set $A$, equipped with the maps $\oplus $, $\ominus $ and $\odot $ and the two elements $\overrightarrow {0}$ and $\overrightarrow {1}$, is a (noncommutative) ring.
The set $A$, equipped with the maps $\oplus $, $\ominus $ and $\rightharpoonup $ and the element $\overrightarrow {0}$, is a $K$-module.
We have

\begin{equation} \lambda \rightharpoonup \left( a\odot b\right) =\left( \lambda \rightharpoonup a\right) \odot b=a\odot \left( \lambda \rightharpoonup b\right) \label{eq.def.alg.Kalg.scaleinv}\end{equation}

for all $\lambda \in K$ and $a,b\in A$.

(Thus, in a nutshell, a $K$-algebra is a set $A$ that is simultaneously a ring and a $K$-module, with the property that the ring $A$ and the $K$-module $A$ have the same addition, the same subtraction and the same zero, and satisfy the additional compatibility property (14).)

docs source

theorem FPS.kalg_add_comm {A : Type u_2} [Ring A] (a b : A) :

a + b = b + a

docs source

theorem FPS.kalg_add_assoc {A : Type u_2} [Ring A] (a b c : A) :

a + (b + c) = a + b + c

docs source

theorem FPS.kalg_add_zero {A : Type u_2} [Ring A] (a : A) :

a + 0 = a

docs source

theorem FPS.kalg_mul_assoc {A : Type u_2} [Ring A] (a b c : A) :

a * (b * c) = a * b * c

docs source

theorem FPS.kalg_left_distrib {A : Type u_2} [Ring A] (a b c : A) :

a * (b + c) = a * b + a * c

docs source

theorem FPS.kalg_right_distrib {A : Type u_2} [Ring A] (a b c : A) :

(a + b) * c = a * c + b * c

docs source

theorem FPS.kalg_mul_one {A : Type u_2} [Ring A] (a : A) :

a * 1 = a

docs source

theorem FPS.kalg_one_mul {A : Type u_2} [Ring A] (a : A) :

1 * a = a

docs source

theorem FPS.kalg_smul_assoc {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (u v : K) (a : A) :

u • v • a = (u * v) • a

docs source

theorem FPS.kalg_smul_add {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (u : K) (a b : A) :

u • (a + b) = u • a + u • b

docs source

theorem FPS.kalg_add_smul {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (u v : K) (a : A) :

(u + v) • a = u • a + v • a

docs source

theorem FPS.kalg_one_smul {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (a : A) :

1 • a = a

docs source

theorem FPS.kalg_zero_smul {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (a : A) :

0 • a = 0

docs source

theorem FPS.kalg_smul_zero {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (u : K) :

u • 0 = 0

docs source

theorem FPS.kalg_smul_mul_assoc {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (c : K) (a b : A) :

c • (a * b) = c • a * b

docs source

theorem FPS.kalg_mul_smul_comm {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (c : K) (a b : A) :

c • (a * b) = a * c • b

docs source

theorem FPS.kalg_smul_mul_eq_mul_smul {K : Type u_1} [CommRing K] {A : Type u_2} [Ring A] [Algebra K A] (c : K) (a b : A) :

c • a * b = a * c • b

Evaluation aka substitution into polynomials

blueprint

Definition 1.153

Let $f\in K\left[ x\right] $ be a polynomial. Let $A$ be any $K$-algebra. Let $a\in A$ be any element. We then define an element $f\left[ a\right] \in A$ as follows:

Write $f$ in the form $f=\sum _{n\in \mathbb {N}}f_{n}x^{n}$ with $f_{0},f_{1},f_{2},\ldots \in K$. (That is, $f_{n}=\left[ x^{n}\right] f$ for each $n\in \mathbb {N}$.) Then, set

\[ f\left[ a\right] :=\sum _{n\in \mathbb {N}}f_{n}a^{n}. \]

(This sum is essentially finite, since $f$ is a polynomial.)

The element $f\left[ a\right] $ is also denoted by $f\circ a$ or by $f\left( a\right) $, and is called the value of $f$ at $a$ (or the evaluation of $f$ at $a$, or the result of substituting $a$ for $x$ in $f$).

docs source

abbrev FPS.polyEval {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (f : Polynomial K) (a : A) :

FPS.polyEval f a = (Polynomial.aeval a) f

blueprint

Theorem 1.157

Let $A$ be a $K$-algebra. Let $a\in A$. Then:

(a) Any $f,g\in K\left[ x\right] $ satisfy

\[ \left( f+g\right) \left[ a\right] =f\left[ a\right] +g\left[ a\right] \ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \left( fg\right) \left[ a\right] =f\left[ a\right] \cdot g\left[ a\right] . \]

(b) Any $\lambda \in K$ and $f\in K\left[ x\right] $ satisfy

\[ \left( \lambda f\right) \left[ a\right] =\lambda \cdot f\left[ a\right] . \]

(c) Any $\lambda \in K$ satisfies $\underline{\lambda }\left[ a\right] =\lambda \cdot 1_{A}$, where $1_{A}$ is the unity of the ring $A$.

(d) We have $x\left[ a\right] =a$.

(e) We have $x^{i}\left[ a\right] =a^{i}$ for each $i\in \mathbb {N}$.

(f) Any $f,g\in K\left[ x\right] $ satisfy $f\left[ g\left[ a\right] \right] =\left( f\left[ g\right] \right) \left[ a\right] $.

docs source

theorem FPS.eval_add' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (f g : Polynomial K) (a : A) :

polyEval (f + g) a = polyEval f a + polyEval g a

docs source

theorem FPS.eval_mul' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (f g : Polynomial K) (a : A) :

polyEval (f * g) a = polyEval f a * polyEval g a

docs source

theorem FPS.eval_smul' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (f : Polynomial K) (c : K) (a : A) :

polyEval (c • f) a = c • polyEval f a

docs source

theorem FPS.eval_C' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (c : K) (a : A) :

polyEval (Polynomial.C c) a = (algebraMap K A) c

docs source

theorem FPS.eval_X' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (a : A) :

polyEval Polynomial.X a = a

docs source

theorem FPS.eval_X_pow' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (a : A) (i : ℕ) :

polyEval (Polynomial.X ^ i) a = a ^ i

docs source

theorem FPS.eval_comp' {K : Type u_1} [CommRing K] {A : Type u_2} [CommRing A] [Algebra K A] (f g : Polynomial K) (a : A) :

polyEval f (polyEval g a) = polyEval (f.comp g) a

Substitution and evaluation of power series

Defining substitution

blueprint

Definition 1.161

Let $f$ and $g$ be two FPSs in $K[[x]]$. Assume that $[x^0]g = 0$ (that is, $g = g_1 x + g_2 x^2 + g_3 x^3 + \cdots $ for some $g_1, g_2, g_3, \ldots \in K$).

We then define an FPS $f[g] \in K[[x]]$ as follows:

Write $f$ in the form $f = \sum _{n \in \mathbb {N}} f_n x^n$ with $f_0, f_1, f_2, \ldots \in K$. (That is, $f_n = [x^n]f$ for each $n \in \mathbb {N}$.) Then, set

\[ f[g] := \sum _{n \in \mathbb {N}} f_n g^n. \]

(This sum is well-defined, as we will see in Proposition 1.163 (b) below.)

This FPS $f[g]$ is also denoted by $f \circ g$, and is called the composition of $f$ with $g$, or the result of substituting $g$ for $x$ in $f$.

Equivalently, the $n$-th coefficient of $f[g]$ is the finite sum

\[ [x^n](f[g]) = \sum _{d=0}^{n} f_d \cdot [x^n](g^d). \]

docs source

noncomputable def AlgebraicCombinatorics.fps_comp {K : Type u_1} [CommRing K] (f g : PowerSeries K) (_hg : PowerSeries.constantCoeff g = 0) :

AlgebraicCombinatorics.fps_comp f g _hg = PowerSeries.subst g f

docs source

theorem AlgebraicCombinatorics.fps_comp_coeff {K : Type u_1} [CommRing K] (f g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (n : ℕ) :

(PowerSeries.coeff n) (PowerSeries.subst g f) = ∑ᶠ (d : ℕ), (PowerSeries.coeff d) f * (PowerSeries.coeff n) (g ^ d)

docs source

theorem AlgebraicCombinatorics.fps_comp_coeff_finite {K : Type u_1} [CommRing K] (f g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (n : ℕ) :

(PowerSeries.coeff n) (PowerSeries.subst g f) = ∑ d ∈ Finset.range (n + 1), (PowerSeries.coeff d) f * (PowerSeries.coeff n) (g ^ d)

blueprint

Proposition 1.163

Let $f$ and $g$ be two FPSs in $K[[x]]$. Assume that $[x^0]g = 0$. Write $f$ in the form $f = \sum _{n \in \mathbb {N}} f_n x^n$ with $f_0, f_1, f_2, \ldots \in K$. Then:

(a) For each $n \in \mathbb {N}$, the first $n$ coefficients of the FPS $g^n$ are $0$.

(b) The sum $\sum _{n \in \mathbb {N}} f_n g^n$ is well-defined, i.e., the family $(f_n g^n)_{n \in \mathbb {N}}$ is summable.

(c) We have $[x^0](\sum _{n \in \mathbb {N}} f_n g^n) = f_0$.

docs source

theorem AlgebraicCombinatorics.fps_subs_wd_firstCoeffs {K : Type u_1} [CommRing K] (g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (n k : ℕ) (hk : k < n) :

(PowerSeries.coeff k) (g ^ n) = 0

docs source

theorem AlgebraicCombinatorics.fps_subs_wd_summable {K : Type u_1} [CommRing K] (_f g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.HasSubst g

docs source

theorem AlgebraicCombinatorics.fps_subs_wd_constCoeff {K : Type u_1} [CommRing K] (f g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.constantCoeff (PowerSeries.subst g f) = PowerSeries.constantCoeff f

Laws of substitution

blueprint

Proposition 1.181

Composition of FPSs satisfies the rules you would expect it to satisfy:

(a) If $f_1, f_2, g \in K[[x]]$ satisfy $[x^0]g = 0$, then $(f_1 + f_2) \circ g = f_1 \circ g + f_2 \circ g$.

(b) If $f_1, f_2, g \in K[[x]]$ satisfy $[x^0]g = 0$, then $(f_1 \cdot f_2) \circ g = (f_1 \circ g) \cdot (f_2 \circ g)$.

(c) If $f_1, f_2, g \in K[[x]]$ satisfy $[x^0]g = 0$, then $\frac{f_1}{f_2} \circ g = \frac{f_1 \circ g}{f_2 \circ g}$, as long as $f_2$ is invertible. (In particular, $f_2 \circ g$ is automatically invertible under these assumptions.)

(d) If $f, g \in K[[x]]$ satisfy $[x^0]g = 0$, then $f^k \circ g = (f \circ g)^k$ for each $k \in \mathbb {N}$.

(e) If $f, g, h \in K[[x]]$ satisfy $[x^0]g = 0$ and $[x^0]h = 0$, then $[x^0](g \circ h) = 0$ and $(f \circ g) \circ h = f \circ (g \circ h)$.

(f) We have $\underline{a} \circ g = \underline{a}$ for each $a \in K$ and $g \in K[[x]]$.

(g) We have $x \circ g = g \circ x = g$ for each $g \in K[[x]]$.

(h) If $(f_i)_{i \in I} \in K[[x]]^I$ is a summable family of FPSs, and if $g \in K[[x]]$ is an FPS satisfying $[x^0]g = 0$, then the family $(f_i \circ g)_{i \in I} \in K[[x]]^I$ is summable as well and we have $(\sum _{i \in I} f_i) \circ g = \sum _{i \in I} f_i \circ g$.

docs source

theorem AlgebraicCombinatorics.fps_subs_add {K : Type u_1} [CommRing K] (f₁ f₂ g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.subst g (f₁ + f₂) = PowerSeries.subst g f₁ + PowerSeries.subst g f₂

docs source

theorem AlgebraicCombinatorics.fps_subs_mul {K : Type u_1} [CommRing K] (f₁ f₂ g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.subst g (f₁ * f₂) = PowerSeries.subst g f₁ * PowerSeries.subst g f₂

docs source

theorem AlgebraicCombinatorics.fps_subs_div {K : Type u_2} [Field K] (f₁ f₂ g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (hf₂ : PowerSeries.constantCoeff f₂ ≠ 0) :

PowerSeries.subst g f₁ * (PowerSeries.subst g f₂)⁻¹ = PowerSeries.subst g (f₁ * f₂⁻¹)

docs source

theorem AlgebraicCombinatorics.fps_subs_pow {K : Type u_1} [CommRing K] (f g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (k : ℕ) :

PowerSeries.subst g (f ^ k) = PowerSeries.subst g f ^ k

docs source

theorem AlgebraicCombinatorics.fps_subs_assoc_constCoeff {K : Type u_1} [CommRing K] (g h : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (hh : PowerSeries.constantCoeff h = 0) :

PowerSeries.constantCoeff (PowerSeries.subst h g) = 0

docs source

theorem AlgebraicCombinatorics.fps_subs_assoc {K : Type u_1} [CommRing K] (f g h : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (hh : PowerSeries.constantCoeff h = 0) :

PowerSeries.subst h (PowerSeries.subst g f) = PowerSeries.subst (PowerSeries.subst h g) f

docs source

theorem AlgebraicCombinatorics.fps_subs_const {K : Type u_1} [CommRing K] (a : K) (g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.subst g (PowerSeries.C a) = PowerSeries.C a

docs source

theorem AlgebraicCombinatorics.fps_subs_X_left {K : Type u_1} [CommRing K] (g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.subst g PowerSeries.X = g

docs source

theorem AlgebraicCombinatorics.fps_subs_X_right {K : Type u_1} [CommRing K] (g : PowerSeries K) :

PowerSeries.subst PowerSeries.X g = g

docs source

theorem AlgebraicCombinatorics.fps_subs_sum {K : Type u_1} [CommRing K] {ι : Type u_2} (s : Finset ι) (f : ι → PowerSeries K) (g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) :

PowerSeries.subst g (∑ i ∈ s, f i) = ∑ i ∈ s, PowerSeries.subst g (f i)

docs source

theorem AlgebraicCombinatorics.fps_subs_summable {K : Type u_1} [CommRing K] {ι : Type u_2} (f : ι → PowerSeries K) (g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (hf : FPS.SummableFPS f) :

FPS.SummableFPS fun (i : ι) => PowerSeries.subst g (f i)

docs source

theorem AlgebraicCombinatorics.fps_subs_summableFPSSum {K : Type u_1} [CommRing K] {ι : Type u_2} (f : ι → PowerSeries K) (g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (hf : FPS.SummableFPS f) :

PowerSeries.subst g (FPS.summableFPSSum f hf) = FPS.summableFPSSum (fun (i : ι) => PowerSeries.subst g (f i)) ⋯

blueprint

Lemma 1.166

Let $f, g \in K[[x]]$ satisfy $[x^0]g = 0$. Let $k \in \mathbb {N}$ be such that the first $k$ coefficients of $f$ are $0$. Then, the first $k$ coefficients of $f \circ g$ are $0$.

docs source

theorem AlgebraicCombinatorics.fps_fg_coeffs_zero {K : Type u_1} [CommRing K] (f g : PowerSeries K) (hg : PowerSeries.constantCoeff g = 0) (k : ℕ) (hf : ∀ m < k, (PowerSeries.coeff m) f = 0) (m : ℕ) :

m < k → (PowerSeries.coeff m) (PowerSeries.subst g f) = 0

blueprint

Definition 1.167

If $i$ and $j$ are any objects, then $\delta _{i,j}$ means the element

\[ \delta _{i,j} = \begin{cases} 1, & \text{if } i = j; \\ 0, & \text{if } i \neq j \end{cases} \]

of $K$.

docs source

def AlgebraicCombinatorics.kroneckerDelta {K : Type u_1} [CommRing K] {α : Type u_2} [DecidableEq α] (i j : α) :

AlgebraicCombinatorics.kroneckerDelta i j = if i = j then 1 else 0

Derivatives of FPSs

blueprint

Definition 1.184

Let $f\in K\left[ \left[ x\right] \right] $ be an FPS. Then, the derivative $f^{\prime }$ of $f$ is an FPS defined as follows: Write $f$ as $f=\sum _{n\in \mathbb {N}}f_{n}x^{n}$ (with $f_{0},f_{1},f_{2},\ldots \in K$), and set

\[ f^{\prime }:=\sum _{n{\gt}0}nf_{n}x^{n-1}. \]

docs source

theorem AlgebraicCombinatorics.FPS.coeff_derivative_eq {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (n : ℕ) :

(PowerSeries.coeff n) ((PowerSeries.derivative R) f) = (PowerSeries.coeff (n + 1)) f * (↑n + 1)

blueprint

Theorem 1.195

(a) We have $\left( f+g\right) ^{\prime }=f^{\prime }+g^{\prime }$ for any $f,g\in K\left[ \left[ x\right] \right] $.

(b) If $\left( f_{i}\right) _{i\in I}$ is a summable family of FPSs, then the family $\left( f_{i}^{\prime }\right) _{i\in I}$ is summable as well, and we have

\[ \left( \sum _{i\in I}f_{i}\right) ^{\prime }=\sum _{i\in I}f_{i}^{\prime }. \]

(c) We have $\left( cf\right) ^{\prime }=cf^{\prime }$ for any $c\in K$ and $f\in K\left[ \left[ x\right] \right] $.

(d) We have $\left( fg\right) ^{\prime }=f^{\prime }g+fg^{\prime }$ for any $f,g\in K\left[ \left[ x\right] \right] $. (This is known as the Leibniz rule.)

(e) If $f,g\in K\left[ \left[ x\right] \right] $ are two FPSs such that $g$ is invertible, then

\[ \left( \frac{f}{g}\right) ^{\prime }=\frac{f^{\prime }g-fg^{\prime }}{g^{2}}. \]

(This is known as the quotient rule.)

(f) If $g\in K\left[ \left[ x\right] \right] $ is an FPS, then $\left( g^{n}\right) ^{\prime }=ng^{\prime }g^{n-1}$ for any $n\in \mathbb {N}$ (where the expression $ng^{\prime }g^{n-1}$ is to be understood as $0$ if $n=0$).

(g) Given two FPSs $f,g\in K\left[ \left[ x\right] \right] $, we have

\[ \left( f\circ g\right) ^{\prime }=\left( f^{\prime }\circ g\right) \cdot g^{\prime } \]

if $f$ is a polynomial or if $\left[ x^{0}\right] g=0$. (This is known as the chain rule.)

(h) If $K$ is a $\mathbb {Q}$-algebra, and if two FPSs $f,g\in K\left[ \left[ x\right] \right] $ satisfy $f^{\prime }=g^{\prime }$, then $f-g$ is constant.

docs source

theorem AlgebraicCombinatorics.FPS.derivative_add {R : Type u_1} [CommSemiring R] (f g : PowerSeries R) :

(PowerSeries.derivative R) (f + g) = (PowerSeries.derivative R) f + (PowerSeries.derivative R) g

docs source

theorem AlgebraicCombinatorics.FPS.derivative_sum {R : Type u_1} [CommSemiring R] {ι : Type u_2} (s : Finset ι) (f : ι → PowerSeries R) :

(PowerSeries.derivative R) (∑ i ∈ s, f i) = ∑ i ∈ s, (PowerSeries.derivative R) (f i)

docs source

theorem AlgebraicCombinatorics.FPS.derivative_summableFPSSum {R' : Type u_2} [CommRing R'] {ι : Type u_3} (f : ι → PowerSeries R') (hf : SummableFPS f) :

(PowerSeries.derivative R') (summableFPSSum f hf) = summableFPSSum (fun (i : ι) => (PowerSeries.derivative R') (f i)) ⋯

docs source

theorem AlgebraicCombinatorics.FPS.derivative_smul {R : Type u_1} [CommSemiring R] (c : R) (f : PowerSeries R) :

(PowerSeries.derivative R) (c • f) = c • (PowerSeries.derivative R) f

docs source

theorem AlgebraicCombinatorics.FPS.derivative_mul {R : Type u_1} [CommSemiring R] (f g : PowerSeries R) :

(PowerSeries.derivative R) (f * g) = (PowerSeries.derivative R) f * g + f * (PowerSeries.derivative R) g

docs source

theorem AlgebraicCombinatorics.FPS.derivative_div {K : Type u_3} [Field K] (f g : PowerSeries K) (hg : PowerSeries.constantCoeff g ≠ 0) :

(PowerSeries.derivative K) (f * g⁻¹) = ((PowerSeries.derivative K) f * g - f * (PowerSeries.derivative K) g) * g⁻¹ ^ 2

docs source

theorem AlgebraicCombinatorics.FPS.derivative_pow' {A : Type u_4} [CommRing A] (g : PowerSeries A) (n : ℕ) :

(PowerSeries.derivative A) (g ^ n) = ↑n * g ^ (n - 1) * (PowerSeries.derivative A) g

docs source

theorem AlgebraicCombinatorics.FPS.derivative_comp {A : Type u_4} [CommRing A] (f g : PowerSeries A) (hg : PowerSeries.HasSubst g) :

(PowerSeries.derivative A) (PowerSeries.subst g f) = PowerSeries.subst g ((PowerSeries.derivative A) f) * (PowerSeries.derivative A) g

docs source

theorem AlgebraicCombinatorics.FPS.derivative_eq_imp_diff_const {R : Type u_5} [CommRing R] [IsAddTorsionFree R] {f g : PowerSeries R} (h : (PowerSeries.derivative R) f = (PowerSeries.derivative R) g) (n : ℕ) :

n ≠ 0 → (PowerSeries.coeff n) (f - g) = 0

Exponentials and logarithms

Definitions

blueprint

Definition 1.211

Define three FPS $\exp $, $\overline{\log }$ and $\overline{\exp }$ in $K\left[\left[x\right]\right]$ by

\begin{align*} \exp & :=\sum _{n\in \mathbb {N}}\frac{1}{n!}x^{n},\\ \overline{\log } & :=\sum _{n\geq 1}\frac{\left(-1\right)^{n-1}}{n}x^{n},\\ \overline{\exp } & :=\exp -1=\sum _{n\geq 1}\frac{1}{n!}x^{n}. \end{align*}

(The last equality sign here follows from $\exp =\sum _{n\in \mathbb {N}}\frac{1}{n!}x^{n}=\underbrace{\frac{1}{0!}}_{=1}\underbrace{x^{0}}_{=1} +\sum _{n\geq 1}\frac{1}{n!}x^{n}=1+\sum _{n\geq 1}\frac{1}{n!}x^{n}$.)

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noncomputable def PowerSeries.logbar (K : Type u_1) [CommRing K] [Algebra ℚ K] :

PowerSeries.logbar K = PowerSeries.mk fun (n : ℕ) => if n = 0 then 0 else (algebraMap ℚ K) ((-1) ^ (n - 1) / ↑n)

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noncomputable def PowerSeries.expbar (K : Type u_1) [CommRing K] [Algebra ℚ K] :

Function.LeftInverse Log Exp ∧ Function.RightInverse Log Exp

PowerSeries.expbar K = PowerSeries.exp K - 1

The exponential and the logarithm are inverse

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Proposition 1.216

Let $g\in K\left[\left[x\right]\right]$ with $\left[x^{0}\right]g=0$. Then:

(a) We have

\[ \left(\overline{\exp }\circ g\right)^{\prime }=\left(\exp \circ g\right) ^{\prime }=\left(\exp \circ g\right)\cdot g^{\prime }. \]

(b) We have

\[ \left(\overline{\log }\circ g\right)^{\prime }=\left(1+g\right) ^{-1}\cdot g^{\prime }. \]

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theorem PowerSeries.derivative_exp_comp {K : Type u_1} [CommRing K] [Algebra ℚ K] {g : PowerSeries K} (hg : constantCoeff g = 0) :

(derivative K) (subst g (exp K)) = subst g (exp K) * (derivative K) g

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theorem PowerSeries.derivative_expbar_comp {K : Type u_1} [CommRing K] [Algebra ℚ K] {g : PowerSeries K} (hg : constantCoeff g = 0) :

(derivative K) (subst g (expbar K)) = subst g (exp K) * (derivative K) g

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theorem PowerSeries.derivative_logbar_comp {K : Type u_2} [Field K] [Algebra ℚ K] {g : PowerSeries K} (hg : constantCoeff g = 0) :

(derivative K) (subst g (logbar K)) = (1 + g)⁻¹ * (derivative K) g

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Lemma 1.220

Let $f,g\in K\left[\left[x\right] \right]$ be two FPSs with $\left[x^{0}\right]g=0$. Then, $\left[ x^{0}\right]\left(f\circ g\right)=\left[x^{0}\right]f$.

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theorem PowerSeries.constantCoeff_subst_of_constantCoeff_zero {K : Type u_1} [CommRing K] {f g : PowerSeries K} (hg : constantCoeff g = 0) :

constantCoeff (subst g f) = constantCoeff f

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Theorem 1.221

We have

\[ \overline{\exp }\circ \overline{\log }=x\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \overline{\log }\circ \overline{\exp }=x. \]

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theorem PowerSeries.expbar_comp_logbar {K : Type u_1} [CommRing K] [Algebra ℚ K] :

subst (logbar K) (expbar K) = X

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theorem PowerSeries.logbar_comp_expbar {K : Type u_1} [CommRing K] [Algebra ℚ K] :

subst (expbar K) (logbar K) = X

The exponential and the logarithm of an FPS

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Definition 1.222

(a) We let $K\left[\left[x\right] \right]_{0}$ denote the set of all FPSs $f\in K\left[\left[x\right] \right]$ with $\left[x^{0}\right]f=0$.

(b) We let $K\left[\left[x\right]\right]_{1}$ denote the set of all FPSs $f\in K\left[\left[x\right]\right]$ with $\left[ x^{0}\right]f=1$.

(c) We define two maps

\begin{align*} \operatorname {Exp}:K\left[\left[x\right]\right]_{0} & \rightarrow K\left[\left[x\right]\right]_{1},\\ g & \mapsto \exp \circ g \end{align*}

and

\begin{align*} \operatorname {Log}:K\left[\left[x\right]\right]_{1} & \rightarrow K\left[\left[x\right]\right]_{0},\\ f & \mapsto \overline{\log }\circ \left(f-1\right). \end{align*}

(These two maps are well-defined according to parts (c) and (d) of Lemma 1.223 below.)

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def PowerSeries.PowerSeries₀ {R : Type u_2} [CommRing R] :

Set (PowerSeries R)

PowerSeries.PowerSeries₀ = {f : PowerSeries R | PowerSeries.constantCoeff f = 0}

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def PowerSeries.PowerSeries₁ {R : Type u_2} [CommRing R] :

Set (PowerSeries R)

PowerSeries.PowerSeries₁ = {f : PowerSeries R | PowerSeries.constantCoeff f = 1}

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noncomputable def PowerSeries.Exp {K : Type u_2} [CommRing K] [Algebra ℚ K] (g : ↑PowerSeries₀) :

↑PowerSeries₁

PowerSeries.Exp g = ⟨PowerSeries.subst (↑g) (PowerSeries.exp K), ⋯⟩

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noncomputable def PowerSeries.Log {K : Type u_2} [CommRing K] [Algebra ℚ K] (f : ↑PowerSeries₁) :

↑PowerSeries₀

PowerSeries.Log f = ⟨PowerSeries.subst (↑f - 1) (PowerSeries.logbar K), ⋯⟩

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Lemma 1.223

(a) For any $f,g\in K\left[\left[ x\right]\right]_{0}$, we have $f\circ g\in K\left[\left[x\right] \right]_{0}$.

(b) For any $f\in K\left[\left[x\right]\right]_{1}$ and $g\in K\left[\left[x\right]\right]_{0}$, we have $f\circ g\in K\left[ \left[x\right]\right]_{1}$.

(c) For any $g\in K\left[\left[x\right]\right]_{0}$, we have $\exp \circ g\in K\left[\left[x\right]\right]_{1}$.

(d) For any $f\in K\left[\left[x\right]\right]_{1}$, we have $f-1\in K\left[\left[x\right]\right]_{0}$ and $\overline{\log } \circ \left(f-1\right)\in K\left[\left[x\right]\right]_{0}$.

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theorem PowerSeries.PowerSeries₀.subst_mem {K : Type u_2} [CommRing K] {f g : PowerSeries K} (hf : f ∈ PowerSeries₀) (hg : g ∈ PowerSeries₀) :

subst g f ∈ PowerSeries₀

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theorem PowerSeries.PowerSeries₁.subst_mem {K : Type u_2} [CommRing K] {f g : PowerSeries K} (hf : f ∈ PowerSeries₁) (hg : g ∈ PowerSeries₀) :

subst g f ∈ PowerSeries₁

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theorem PowerSeries.exp_subst_mem_PowerSeries₁ {K : Type u_2} [CommRing K] [Algebra ℚ K] {g : PowerSeries K} (hg : g ∈ PowerSeries₀) :

subst g (exp K) ∈ PowerSeries₁

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theorem PowerSeries.logbar_subst_sub_one_mem_PowerSeries₀ {K : Type u_2} [CommRing K] [Algebra ℚ K] {f : PowerSeries K} (hf : f ∈ PowerSeries₁) :

subst (f - 1) (logbar K) ∈ PowerSeries₀

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Lemma 1.224

The maps $\operatorname {Exp}$ and $\operatorname {Log}$ are mutually inverse bijections between $K\left[ \left[x\right]\right]_{0}$ and $K\left[\left[x\right]\right]_{1}$.

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theorem PowerSeries.Log_Exp {K : Type u_2} [CommRing K] [Algebra ℚ K] (g : ↑PowerSeries₀) :

Log (Exp g) = g

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theorem PowerSeries.Exp_Log {K : Type u_2} [CommRing K] [Algebra ℚ K] (f : ↑PowerSeries₁) :

Exp (Log f) = f

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theorem PowerSeries.Exp_Log_inverse {K : Type u_2} [CommRing K] [Algebra ℚ K] :

Addition to multiplication

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Lemma 1.228

(a) For any $f,g\in K\left[\left[ x\right]\right]_{0}$, we have

\[ \operatorname {Exp}\left(f+g\right)=\operatorname {Exp}f\cdot \operatorname {Exp}g. \]

(b) For any $f,g\in K\left[\left[x\right]\right]_{1}$, we have

\[ \operatorname {Log}\left(fg\right)=\operatorname {Log} f+\operatorname {Log}g. \]

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theorem PowerSeries.Exp_add {K : Type u_2} [CommRing K] [Algebra ℚ K] (f g : ↑PowerSeries₀) :

Exp ⟨↑f + ↑g, ⋯⟩ = ⟨↑(Exp f) * ↑(Exp g), ⋯⟩

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theorem PowerSeries.Log_mul {K : Type u_2} [CommRing K] [Algebra ℚ K] (f g : ↑PowerSeries₁) :

Log ⟨↑f * ↑g, ⋯⟩ = ⟨↑(Log f) + ↑(Log g), ⋯⟩

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Proposition 1.229

(a) The subset $K\left[\left[ x\right]\right]_{0}$ of $K\left[\left[x\right]\right]$ is closed under addition and subtraction and contains $0$, and thus forms a group $\left(K\left[\left[x\right]\right]_{0},+,0\right)$.

(b) The subset $K\left[\left[x\right]\right]_{1}$ of $K\left[\left[x\right]\right]$ is closed under multiplication and division and contains $1$, and thus forms a group $\left(K\left[\left[ x\right]\right]_{1},\cdot ,1\right)$.

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def PowerSeries.PowerSeries₀.addSubgroup {R : Type u_2} [CommRing R] :

AddSubgroup (PowerSeries R)

PowerSeries.PowerSeries₀.addSubgroup = { carrier := PowerSeries.PowerSeries₀, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

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def PowerSeries.PowerSeries₁.subgroup {R : Type u_2} [Field R] :

Subgroup (PowerSeries R)ˣ

PowerSeries.PowerSeries₁.subgroup = { carrier := {u : (PowerSeries R)ˣ | ↑u ∈ PowerSeries.PowerSeries₁ }, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }

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Theorem 1.230

The maps

\[ \operatorname {Exp}:\left(K\left[\left[x\right]\right]_{0} ,+,0\right)\rightarrow \left(K\left[\left[x\right]\right]_{1} ,\cdot ,1\right) \]

and

\[ \operatorname {Log}:\left(K\left[\left[x\right]\right]_{1} ,\cdot ,1\right)\rightarrow \left(K\left[\left[x\right]\right] _{0},+,0\right) \]

are mutually inverse group isomorphisms.

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theorem PowerSeries.Exp_Log_groupIso {K : Type u_2} [CommRing K] [Algebra ℚ K] :

(Function.LeftInverse Log Exp ∧ Function.RightInverse Log Exp) ∧ (∀ (f g : ↑PowerSeries₀), Exp ⟨↑f + ↑g, ⋯⟩ = ⟨↑(Exp f) * ↑(Exp g), ⋯⟩) ∧ Exp ⟨0, ⋯⟩ = ⟨1, ⋯⟩

The logarithmic derivative

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Definition 1.233

In this definition, we do not use Convention 1.210; thus, $K$ can be an arbitrary commutative ring. However, we set $K\left[\left[x\right]\right]_{1}=\left\{ f\in K\left[\left[x\right]\right]\ \mid \ \left[x^{0}\right]f=1\right\} $.

For any FPS $f\in K\left[\left[x\right]\right]_{1}$, we define the logarithmic derivative $\operatorname {loder}f\in K\left[\left[ x\right]\right]$ to be the FPS $\frac{f^{\prime }}{f}$. (This is well-defined, since $f$ is easily seen to be invertible.)

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noncomputable def PowerSeries.loder {R : Type u_2} [Field R] (f : PowerSeries R) :

PowerSeries R

f.loder = (PowerSeries.derivative R) f * f⁻¹

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Proposition 1.236

Let $K$ be a commutative $\mathbb {Q}$-algebra. Let $f\in K\left[\left[x\right]\right]_{1}$ be an FPS. Then, $\operatorname {loder}f=\left(\operatorname {Log}f\right)^{\prime }$.

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theorem PowerSeries.loder_eq_derivative_Log {R : Type u_2} [Field R] [Algebra ℚ R] {f : PowerSeries R} (hf : constantCoeff f = 1) :

f.loder = (derivative R) (subst (f - 1) (logbar R))

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Proposition 1.237

Let $f,g\in K\left[\left[x\right]\right] _{1}$ be two FPSs. Then, $\operatorname {loder}\left(fg\right) =\operatorname {loder}f+\operatorname {loder}g$.

(Here, we do not use Convention 1.210; thus, $K$ can be an arbitrary commutative ring.)

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theorem PowerSeries.loder_mul {R : Type u_2} [Field R] {f g : PowerSeries R} (hf : constantCoeff f = 1) (hg : constantCoeff g = 1) :

(f * g).loder = f.loder + g.loder

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Corollary 1.238

Let $f_{1},f_{2},\ldots ,f_{k}$ be any $k$ FPSs in $K\left[\left[x\right]\right]_{1}$. Then,

\[ \operatorname {loder}\left(f_{1}f_{2}\cdots f_{k}\right) =\operatorname {loder}\left(f_{1}\right)+\operatorname {loder}\left( f_{2}\right)+\cdots +\operatorname {loder}\left(f_{k}\right). \]

(Here, we do not use Convention 1.210; thus, $K$ can be an arbitrary commutative ring.)

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theorem PowerSeries.loder_prod {R : Type u_2} [Field R] {ι : Type u_3} (s : Finset ι) (f : ι → PowerSeries R) (hf : ∀ i ∈ s, constantCoeff (f i) = 1) :

(∏ i ∈ s, f i).loder = ∑ i ∈ s, (f i).loder

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Corollary 1.239

Let $f$ be any FPS in $K\left[\left[x\right] \right]_{1}$. Then, $\operatorname {loder}\left(f^{-1}\right) =-\operatorname {loder}f$.

(Here, we do not use Convention 1.210; thus, $K$ can be an arbitrary commutative ring.)

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theorem PowerSeries.loder_inv {R : Type u_2} [Field R] {f : PowerSeries R} (hf : constantCoeff f = 1) :

f⁻¹.loder = -f.loder

Non-integer powers

Definition

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Definition 1.258

Assume that $K$ is a commutative $\mathbb {Q}$-algebra. Let $f\in K\left[ \left[ x\right] \right] _{1}$ and $c\in K$. Then, we define an FPS

\[ f^{c}:=\operatorname {Exp}\left( c\operatorname {Log}f\right) \in K\left[ \left[ x\right] \right] _{1}. \]

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noncomputable def AlgebraicCombinatorics.FPS.fpsPow {K : Type u_1} [CommRing K] [Algebra ℚ K] (f : PowerSeries K) (c : K) :

AlgebraicCombinatorics.FPS.fpsPow f c = AlgebraicCombinatorics.FPS.fpsExp (c • AlgebraicCombinatorics.FPS.fpsLog f)

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Theorem 1.262

Assume that $K$ is a commutative $\mathbb {Q}$-algebra. For any $a,b\in K$ and $f,g\in K\left[ \left[ x\right] \right] _{1}$, we have

\[ f^{a+b}=f^{a}f^{b},\ \ \ \ \ \ \ \ \ \ \left( fg\right) ^{a}=f^{a}g^{a},\ \ \ \ \ \ \ \ \ \ \left( f^{a}\right) ^{b}=f^{ab}. \]

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theorem AlgebraicCombinatorics.FPS.fpsPow_add {K : Type u_1} [CommRing K] [Algebra ℚ K] {f : PowerSeries K} (hf : HasConstantTermOne f) (a b : K) :

fpsPow f (a + b) = fpsPow f a * fpsPow f b

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theorem AlgebraicCombinatorics.FPS.fpsPow_mul {K : Type u_1} [CommRing K] [Algebra ℚ K] {f g : PowerSeries K} (hf : HasConstantTermOne f) (hg : HasConstantTermOne g) (a : K) :

fpsPow (f * g) a = fpsPow f a * fpsPow g a

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theorem AlgebraicCombinatorics.FPS.fpsPow_pow {K : Type u_1} [CommRing K] [Algebra ℚ K] {f : PowerSeries K} (hf : HasConstantTermOne f) (a b : K) :

fpsPow (fpsPow f a) b = fpsPow f (a * b)

The Newton binomial formula for arbitrary exponents

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Theorem 1.264 Generalized Newton binomial formula

Assume that $K$ is a commutative $\mathbb {Q}$-algebra. Let $c\in K$. Then,

\[ \left( 1+x\right) ^{c}=\sum _{k\in \mathbb {N}}\binom {c}{k}x^{k}. \]

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theorem AlgebraicCombinatorics.FPS.generalizedNewtonBinomial {K : Type u_1} [CommRing K] [Algebra ℚ K] [BinomialRing K] [CharZero K] (c : K) :

fpsPow (1 + PowerSeries.X) c = PowerSeries.binomialSeries K c

Another application

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Proposition 1.271

Let $n\in \mathbb {C}$ and $k\in \mathbb {N}$. Then,

\[ \sum _{i=0}^{k}\binom {n+i-1}{i}\binom {n}{k-2i}=\binom {n+k-1}{k}. \]

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theorem AlgebraicCombinatorics.FPS.binomialIdentity {K : Type u_1} [CommRing K] [Algebra ℚ K] [BinomialRing K] [CharZero K] (n : K) (k : ℕ) :

∑ i ∈ Finset.range (k / 2 + 1), Ring.choose (n + ↑i - 1) i * Ring.choose n (k - 2 * i) = Ring.choose (n + ↑k - 1) k

Integer compositions

Compositions

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Definition 1.291

(a) An (integer) composition means a (finite) tuple of positive integers.

(b) The size of an integer composition $\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}\right) $ is defined to be the integer $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{m}$. It is written $\left\vert \alpha \right\vert $.

(c) The length of an integer composition $\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}\right) $ is defined to be the integer $m$. It is written $\ell \left( \alpha \right) $.

(d) Let $n\in \mathbb {N}$. A composition of $n$ means a composition whose size is $n$.

(e) Let $n\in \mathbb {N}$ and $k\in \mathbb {N}$. A composition of $n$ into $k$ parts is a composition whose size is $n$ and whose length is $k$.

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def AlgebraicCombinatorics.Composition :

AlgebraicCombinatorics.Composition = List ℕ+

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def AlgebraicCombinatorics.Composition.size (α : Composition) :

α.size = (List.map (fun (x : ℕ+) => ↑x) α).sum

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def AlgebraicCombinatorics.Composition.len (α : Composition) :

α.len = List.length α

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def AlgebraicCombinatorics.Composition.ofSize (n : ℕ) :

AlgebraicCombinatorics.Composition.ofSize n = { α : AlgebraicCombinatorics.Composition // α.size = n }

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def AlgebraicCombinatorics.Composition.ofSizeIntoParts (n k : ℕ) :

AlgebraicCombinatorics.Composition.ofSizeIntoParts n k = { α : AlgebraicCombinatorics.Composition // α.size = n ∧ α.len = k }

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Theorem 1.304

Let $n,k\in \mathbb {N}$. Then, the # of compositions of $n$ into $k$ parts is

\[ \binom {n-1}{n-k}=\begin{cases} \binom {n-1}{k-1}, & \text{if }n{\gt}0;\\ \delta _{k,0}, & \text{if }n=0. \end{cases} \]

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theorem AlgebraicCombinatorics.Composition.card_ofSizeIntoParts_pos (n k : ℕ) (hn : 0 < n) (hk : 0 < k) :

Fintype.card (ofSizeIntoParts n k) = (n - 1).choose (k - 1)

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theorem AlgebraicCombinatorics.Composition.card_ofSizeIntoParts_k_zero (n : ℕ) (hn : 0 < n) :

Fintype.card (ofSizeIntoParts n 0) = 0

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theorem AlgebraicCombinatorics.Composition.card_ofSizeIntoParts_zero (k : ℕ) :

Fintype.card (ofSizeIntoParts 0 k) = if k = 0 then 1 else 0

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Theorem 1.305

Let $n\in \mathbb {N}$. Then, the # of compositions of $n$ is

\[ \begin{cases} 2^{n-1}, & \text{if }n{\gt}0;\\ 1, & \text{if }n=0. \end{cases} \]

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theorem AlgebraicCombinatorics.Composition.card_ofSize (n : ℕ) :

Fintype.card (ofSize n) = if n = 0 then 1 else 2 ^ (n - 1)

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theorem AlgebraicCombinatorics.Composition.card_ofSize_pos (n : ℕ) (hn : 0 < n) :

Fintype.card (ofSize n) = 2 ^ (n - 1)

Weak compositions

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Definition 1.306

(a) An (integer) weak composition means a (finite) tuple of nonnegative integers.

(b) The size of a weak composition $\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}\right) $ is defined to be the integer $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{m}$. It is written $\left\vert \alpha \right\vert $.

(c) The length of a weak composition $\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}\right) $ is defined to be the integer $m$. It is written $\ell \left( \alpha \right) $.

(d) Let $n\in \mathbb {N}$. A weak composition of $n$ means a weak composition whose size is $n$.

(e) Let $n\in \mathbb {N}$ and $k\in \mathbb {N}$. A weak composition of $n$ into $k$ parts is a weak composition whose size is $n$ and whose length is $k$.

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def AlgebraicCombinatorics.WeakComposition :

AlgebraicCombinatorics.WeakComposition = List ℕ

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def AlgebraicCombinatorics.WeakComposition.size (α : WeakComposition) :

α.size = List.sum α

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def AlgebraicCombinatorics.WeakComposition.len (α : WeakComposition) :

α.len = List.length α

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def AlgebraicCombinatorics.WeakComposition.ofSize (n : ℕ) :

AlgebraicCombinatorics.WeakComposition.ofSize n = { α : AlgebraicCombinatorics.WeakComposition // α.size = n }

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def AlgebraicCombinatorics.WeakComposition.ofSizeIntoParts (n k : ℕ) :

AlgebraicCombinatorics.WeakComposition.ofSizeIntoParts n k = { α : AlgebraicCombinatorics.WeakComposition // α.size = n ∧ α.len = k }

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Theorem 1.311

Let $n,k\in \mathbb {N}$. Then, the # of weak compositions of $n$ into $k$ parts is

\[ \binom {n+k-1}{n}=\begin{cases} \binom {n+k-1}{k-1}, & \text{if }k{\gt}0;\\ \delta _{n,0}, & \text{if }k=0. \end{cases} \]

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theorem AlgebraicCombinatorics.WeakComposition.card_ofSizeIntoParts (n k : ℕ) :

Fintype.card (ofSizeIntoParts n k) = (n + k - 1).choose n

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theorem AlgebraicCombinatorics.WeakComposition.card_ofSizeIntoParts_pos (n k : ℕ) (hk : 0 < k) :

Fintype.card (ofSizeIntoParts n k) = (n + k - 1).choose (k - 1)

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theorem AlgebraicCombinatorics.WeakComposition.card_ofSizeIntoParts_zero (n : ℕ) :

Fintype.card (ofSizeIntoParts n 0) = if n = 0 then 1 else 0

Weak compositions with entries from $\left\{ 0,1,\ldots ,p-1\right\} $

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Theorem 1.319

Let $n,k,p\in \mathbb {N}$. Then, the # of $k$-tuples $\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{k}\right) \in \left\{ 0,1,\ldots ,p-1\right\} ^{k}$ satisfying $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}=n$ is

\[ \sum _{j=0}^{k}\left( -1\right) ^{j}\binom {k}{j}\binom {n-pj+k-1}{n-pj}. \]

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theorem AlgebraicCombinatorics.BoundedWeakComposition.card (n k p : ℕ) :

↑(Fintype.card (BoundedWeakComposition n k p)) = ∑ j ∈ Finset.range (k + 1) with p * j ≤ n, (-1) ^ j * ↑(k.choose j) * ↑((n - p * j + k - 1).choose (n - p * j))

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Proposition 1.321

Let $n,k\in \mathbb {N}$. Then,

\[ \binom {k}{n}=\sum _{j=0}^{k}\left( -1\right) ^{j}\binom {k}{j}\binom {n-2j+k-1}{n-2j}. \]

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theorem AlgebraicCombinatorics.binom_sum_identity (n k : ℕ) :

↑(k.choose n) = ∑ j ∈ Finset.range (k + 1) with 2 * j ≤ n, (-1) ^ j * ↑(k.choose j) * ↑((n - 2 * j + k - 1).choose (n - 2 * j))

$x^{n}$-equivalence

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Definition 1.322

Let $n\in \mathbb {N}$. Let $f,g\in K\left[\left[x\right]\right]$ be two FPSs. We write $f\overset {x^{n}}{\equiv }g$ if and only if

\[ \text{each }m\in \left\{ 0,1,\ldots ,n\right\} \text{ satisfies } \left[x^{m}\right]f=\left[x^{m}\right]g. \]

Thus, we have defined a binary relation $\overset {x^{n}}{\equiv }$ on the set $K\left[\left[x\right]\right]$. We say that an FPS $f$ is $x^{n}$-equivalent to an FPS $g$ if and only if $f\overset {x^{n}}{\equiv }g$.

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def PowerSeries.XnEquiv {R : Type u_1} [CommSemiring R] (n : ℕ) (f g : PowerSeries R) :

(f ≡[x^n] g) = ∀ m ≤ n, (PowerSeries.coeff m) f = (PowerSeries.coeff m) g

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Theorem 1.328

Let $n\in \mathbb {N}$.

(a) The relation $\overset {x^{n}}{\equiv }$ on $K\left[\left[ x\right]\right]$ is an equivalence relation. In other words:

This relation is reflexive (i.e., we have $f\overset {x^{n}}{\equiv }f$ for each $f\in K\left[\left[x\right]\right]$).
This relation is transitive (i.e., if three FPSs $f,g,h\in K\left[ \left[x\right]\right]$ satisfy $f\overset {x^{n}}{\equiv }g$ and $g\overset {x^{n}}{\equiv }h$, then $f\overset {x^{n}}{\equiv }h$).
This relation is symmetric (i.e., if two FPSs $f,g\in K\left[\left[ x\right]\right]$ satisfy $f\overset {x^{n}}{\equiv }g$, then $g\overset {x^{n}}{\equiv }f$).

(b) If $a,b,c,d\in K\left[\left[x\right]\right]$ are four FPSs satisfying $a\overset {x^{n}}{\equiv }b$ and $c\overset {x^{n}}{\equiv }d$, then we also have

\begin{align} & a+c\overset {x^{n}}{\equiv }b+d;\label{eq.thm.fps.xneq.props.b.+}\\ & a-c\overset {x^{n}}{\equiv }b-d;\label{eq.thm.fps.xneq.props.b.-}\\ & ac\overset {x^{n}}{\equiv }bd. \label{eq.thm.fps.xneq.props.b.*} \end{align}

(c) If $a,b\in K\left[\left[x\right]\right]$ are two FPSs satisfying $a\overset {x^{n}}{\equiv }b$, then $\lambda a\overset {x^{n}}{\equiv }\lambda b$ for each $\lambda \in K$.

(d) If $a,b\in K\left[\left[x\right]\right]$ are two invertible FPSs satisfying $a\overset {x^{n}}{\equiv }b$, then $a^{-1} \overset {x^{n}}{\equiv }b^{-1}$.

(e) If $a,b,c,d\in K\left[\left[x\right]\right]$ are four FPSs satisfying $a\overset {x^{n}}{\equiv }b$ and $c\overset {x^{n}}{\equiv }d$, and if the FPSs $c$ and $d$ are invertible, then we also have

\begin{equation} \frac{a}{c}\overset {x^{n}}{\equiv }\frac{b}{d}. \label{eq.thm.fps.xneq.props.e./} \end{equation}

(f) Let $S$ be a finite set. Let $\left(a_{s}\right)_{s\in S}\in K\left[\left[x\right]\right]^{S}$ and $\left(b_{s}\right)_{s\in S}\in K\left[\left[x\right]\right]^{S}$ be two families of FPSs such that

\begin{equation} \text{each }s\in S\text{ satisfies }a_{s}\overset {x^{n}}{\equiv }b_{s}. \label{eq.thm.fps.xneq.props.e.ass} \end{equation}

Then, we have

\begin{align} & \sum _{s\in S}a_{s}\overset {x^{n}}{\equiv }\sum _{s\in S}b_{s} ;\label{eq.thm.fps.xneq.props.e.+}\\ & \prod _{s\in S}a_{s}\overset {x^{n}}{\equiv }\prod _{s\in S}b_{s}. \label{eq.thm.fps.xneq.props.e.*} \end{align}

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theorem PowerSeries.XnEquiv.equivalence {R : Type u_1} [CommSemiring R] (n : ℕ) :

Equivalence (XnEquiv n)

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theorem PowerSeries.XnEquiv.add {R : Type u_1} [CommSemiring R] {n : ℕ} {a b c d : PowerSeries R} (hab : a ≡[x^n] b) (hcd : c ≡[x^n] d) :

a + c ≡[x^n] b + d

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theorem PowerSeries.XnEquiv.sub {R : Type u_2} [CommRing R] {n : ℕ} {a b c d : PowerSeries R} (hab : a ≡[x^n] b) (hcd : c ≡[x^n] d) :

a - c ≡[x^n] b - d

docs source

theorem PowerSeries.XnEquiv.mul {R : Type u_1} [CommSemiring R] {n : ℕ} {a b c d : PowerSeries R} (hab : a ≡[x^n] b) (hcd : c ≡[x^n] d) :

a * c ≡[x^n] b * d

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theorem PowerSeries.XnEquiv.smul {R : Type u_1} [CommSemiring R] {n : ℕ} {a b : PowerSeries R} (hab : a ≡[x^n] b) (r : R) :

r • a ≡[x^n] r • b

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theorem PowerSeries.XnEquiv.invOfUnit {R : Type u_2} [CommRing R] {n : ℕ} {a b : PowerSeries R} (ua ub : Rˣ) (ha : constantCoeff a = ↑ua) (hb : constantCoeff b = ↑ub) (hab : a ≡[x^n] b) :

a.invOfUnit ua ≡[x^n] b.invOfUnit ub

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theorem PowerSeries.XnEquiv.inv {K : Type u_2} [Field K] {n : ℕ} {a b : PowerSeries K} (ha : constantCoeff a ≠ 0) (hb : constantCoeff b ≠ 0) (hab : a ≡[x^n] b) :

a⁻¹ ≡[x^n] b⁻¹

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theorem PowerSeries.XnEquiv.div {R : Type u_2} [CommRing R] {n : ℕ} {a b c d : PowerSeries R} (hab : a ≡[x^n] b) (hcd : c ≡[x^n] d) (u : Rˣ) (hu : constantCoeff c = ↑u) (v : Rˣ) (hv : constantCoeff d = ↑v) :

a * c.invOfUnit u ≡[x^n] b * d.invOfUnit v

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theorem PowerSeries.XnEquiv.sum {R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] {n : ℕ} {s : Finset ι} {a b : ι → PowerSeries R} (h : ∀ i ∈ s, a i ≡[x^n] b i) :

∑ i ∈ s, a i ≡[x^n] ∑ i ∈ s, b i

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theorem PowerSeries.XnEquiv.prod {R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] {n : ℕ} {s : Finset ι} {a b : ι → PowerSeries R} (h : ∀ i ∈ s, a i ≡[x^n] b i) :

∏ i ∈ s, a i ≡[x^n] ∏ i ∈ s, b i

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Proposition 1.335

Let $n\in \mathbb {N}$. Let $f,g\in K\left[\left[x\right]\right]$ be two FPSs. Then, we have $f\overset {x^{n}}{\equiv }g$ if and only if the FPS $f-g$ is a multiple of $x^{n+1}$.

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theorem PowerSeries.xnEquiv_iff_dvd {R : Type u_3} [CommRing R] {n : ℕ} {f g : PowerSeries R} :

(f ≡[x^n] g) ↔ X ^ (n + 1) ∣ f - g

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Proposition 1.338

Let $n\in \mathbb {N}$. Let $a,b,c,d\in K\left[\left[x\right]\right]$ be four FPSs satisfying $a\overset {x^{n}}{\equiv }b$ and $c\overset {x^{n}}{\equiv }d$ and $[x^{0}]c=0$ and $[x^{0}]d=0$. Then,

\[ a\circ c\overset {x^{n}}{\equiv }b\circ d. \]

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theorem PowerSeries.XnEquiv.comp {R : Type u_3} [CommRing R] {n : ℕ} {a b c d : PowerSeries R} (hab : a ≡[x^n] b) (hcd : c ≡[x^n] d) (hc : constantCoeff c = 0) (hd : constantCoeff d = 0) :

subst c a ≡[x^n] subst d b

Infinite products

A rigorous definition

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Definition 1.340

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a (possibly infinite) family of FPSs. Let $n\in \mathbb {N}$. Let $M$ be a finite subset of $I$.

(a) We say that $M$ determines the $x^{n}$-coefficient in the sum of $\left(\mathbf{a}_{i}\right)_{i\in I}$ if every finite subset $J$ of $I$ satisfying $M\subseteq J\subseteq I$ satisfies

\[ \left[x^{n}\right]\left(\sum _{i\in J}\mathbf{a}_{i}\right) =\left[x^{n}\right]\left(\sum _{i\in M}\mathbf{a}_{i}\right). \]

(b) We say that $M$ determines the $x^{n}$-coefficient in the product of $\left(\mathbf{a}_{i}\right)_{i\in I}$ if every finite subset $J$ of $I$ satisfying $M\subseteq J\subseteq I$ satisfies

\[ \left[x^{n}\right]\left(\prod _{i\in J}\mathbf{a}_{i}\right) =\left[x^{n}\right]\left(\prod _{i\in M}\mathbf{a}_{i}\right). \]

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def PowerSeries.DeterminesCoeffInProd {R : Type u_1} [CommRing R] {I : Type u_2} (a : I → PowerSeries R) (M : Finset I) (n : ℕ) :

PowerSeries.DeterminesCoeffInProd a M n = ∀ (J : Finset I), M ⊆ J → (PowerSeries.coeff n) (∏ i ∈ J, a i) = (PowerSeries.coeff n) (∏ i ∈ M, a i)

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abbrev AlgebraicCombinatorics.FPS.DeterminesCoeff {K : Type u_1} [CommRing K] {ι : Type u_2} (a : ι → PowerSeries K) (n : ℕ) (U : Finset ι) :

AlgebraicCombinatorics.FPS.DeterminesCoeff a n U = PowerSeries.DeterminesCoeffInProd a U n

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Definition 1.341

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a (possibly infinite) family of FPSs. Let $n\in \mathbb {N}$.

(a) We say that the $x^{n}$-coefficient in the sum of $\left(\mathbf{a}_{i}\right)_{i\in I}$ is finitely determined if there is a finite subset $M$ of $I$ that determines the $x^{n}$-coefficient in the sum of $\left(\mathbf{a}_{i}\right)_{i\in I}$.

(b) We say that the $x^{n}$-coefficient in the product of $\left(\mathbf{a}_{i}\right)_{i\in I}$ is finitely determined if there is a finite subset $M$ of $I$ that determines the $x^{n}$-coefficient in the product of $\left(\mathbf{a}_{i}\right)_{i\in I}$.

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def PowerSeries.CoeffFinitelyDeterminedInProd {R : Type u_1} [CommRing R] {I : Type u_2} (a : I → PowerSeries R) (n : ℕ) :

PowerSeries.CoeffFinitelyDeterminedInProd a n = ∃ (M : Finset I), PowerSeries.DeterminesCoeffInProd a M n

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abbrev AlgebraicCombinatorics.FPS.CoeffFinitelyDetermined {K : Type u_1} [CommRing K] {ι : Type u_2} (a : ι → PowerSeries K) (n : ℕ) :

AlgebraicCombinatorics.FPS.CoeffFinitelyDetermined a n = PowerSeries.CoeffFinitelyDeterminedInProd a n

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Proposition 1.342

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a (possibly infinite) family of FPSs. Then:

(a) The family $\left(\mathbf{a}_{i}\right)_{i\in I}$ is summable if and only if each coefficient in its sum is finitely determined (i.e., for each $n\in \mathbb {N}$, the $x^{n}$-coefficient in the sum of $\left(\mathbf{a}_{i}\right)_{i\in I}$ is finitely determined).

(b) If the family $\left(\mathbf{a}_{i}\right)_{i\in I}$ is summable, then its sum $\sum _{i\in I}\mathbf{a}_{i}$ is the FPS whose $x^{n}$-coefficient (for any $n\in \mathbb {N}$) can be computed as follows: If $n\in \mathbb {N}$, and if $M$ is a finite subset of $I$ that determines the $x^{n}$-coefficient in the sum of $\left(\mathbf{a}_{i}\right)_{i\in I}$, then

\[ \left[x^{n}\right]\left(\sum _{i\in I}\mathbf{a}_{i}\right) =\left[x^{n}\right]\left(\sum _{i\in M}\mathbf{a}_{i}\right). \]

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theorem PowerSeries.summable_iff_coeff_finitely_determined {R : Type u_1} [CommRing R] {I : Type u_2} (a : I → PowerSeries R) :

(∀ (n : ℕ), {i : I | (coeff n) (a i) ≠ 0}.Finite) ↔ ∀ (n : ℕ), CoeffFinitelyDeterminedInSum a n

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Definition 1.343

Let $\left(\mathbf{a}_{i}\right)_{i\in I}$ be a (possibly infinite) family of FPSs. Then:

(a) The family $\left(\mathbf{a}_{i}\right)_{i\in I}$ is said to be multipliable if and only if each coefficient in its product is finitely determined.

(b) If the family $\left(\mathbf{a}_{i}\right)_{i\in I}$ is multipliable, then its product $\prod _{i\in I}\mathbf{a}_{i}$ is defined to be the FPS whose $x^{n}$-coefficient (for any $n\in \mathbb {N}$) can be computed as follows: If $n\in \mathbb {N}$, and if $M$ is a finite subset of $I$ that determines the $x^{n}$-coefficient in the product of $\left( \mathbf{a}_{i}\right)_{i\in I}$, then

\[ \left[x^{n}\right]\left(\prod _{i\in I}\mathbf{a}_{i}\right) =\left[x^{n}\right]\left(\prod _{i\in M}\mathbf{a}_{i}\right). \]

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def PowerSeries.Multipliable {R : Type u_1} [CommRing R] {I : Type u_2} (a : I → PowerSeries R) :

PowerSeries.Multipliable a = ∀ (n : ℕ), PowerSeries.CoeffFinitelyDeterminedInProd a n

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abbrev AlgebraicCombinatorics.FPS.Multipliable {K : Type u_1} [CommRing K] {ι : Type u_2} (a : ι → PowerSeries K) :

AlgebraicCombinatorics.FPS.Multipliable a = PowerSeries.Multipliable a

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Proposition 1.344

This definition of $\prod _{i\in I}\mathbf{a}_{i}$ is well-defined – i.e., the coefficient $\left[x^{n}\right]\left(\prod _{i\in M}\mathbf{a}_{i}\right)$ does not depend on $M$ (as long as $M$ is a finite subset of $I$ that determines the $x^{n}$-coefficient in the product of $\left(\mathbf{a}_{i}\right)_{i\in I}$).

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theorem PowerSeries.multipliable_coeff_eq_of_determines {R : Type u_1} [CommRing R] {I : Type u_2} {a : I → PowerSeries R} {M₁ M₂ : Finset I} {n : ℕ} (h₁ : DeterminesCoeffInProd a M₁ n) (h₂ : DeterminesCoeffInProd a M₂ n) :

(coeff n) (∏ i ∈ M₁, a i) = (coeff n) (∏ i ∈ M₂, a i)

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Proposition 1.345

Let $\left(\mathbf{a}_{i}\right)_{i\in I}$ be a finite family of FPSs. Then, the product $\prod _{i\in I}\mathbf{a}_{i}$ defined according to Definition 1.343 (b) equals the finite product $\prod _{i\in I}\mathbf{a}_{i}$ defined in the usual way (i.e., defined as in any commutative ring).

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theorem PowerSeries.tprod_eq_finprod {R : Type u_1} [CommRing R] {I : Type u_2} [Fintype I] (a : I → PowerSeries R) (ha : Multipliable a) :

tprod a ha = ∏ i : I, a i

A general criterion for multipliability

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Lemma 1.348

Let $a,f\in K\left[\left[x\right]\right]$ be two FPSs. Let $n\in \mathbb {N}$. Assume that

\[ \left[x^{m}\right]f=0 \qquad \text{for each }m\in \left\{ 0,1,\ldots ,n\right\} . \]

Then,

\[ \left[x^{m}\right]\left(a\left(1+f\right)\right)=\left[x^{m}\right]a \qquad \text{for each }m\in \left\{ 0,1,\ldots ,n\right\} . \]

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theorem PowerSeries.coeff_mul_one_add_eq_of_coeff_zero {R : Type u_1} [CommRing R] {a f : PowerSeries R} {n : ℕ} (hf : ∀ m ≤ n, (coeff m) f = 0) (m : ℕ) (hm : m ≤ n) :

(coeff m) (a * (1 + f)) = (coeff m) a

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Lemma 1.349

Let $a\in K\left[\left[x\right]\right]$ be an FPS. Let $\left(f_{i}\right)_{i\in J} \in K\left[\left[x\right]\right]^{J}$ be a finite family of FPSs. Let $n\in \mathbb {N}$. Assume that each $i\in J$ satisfies

\[ \left[x^{m}\right]\left(f_{i}\right)=0 \qquad \text{for each } m\in \left\{ 0,1,\ldots ,n\right\} . \]

Then,

\[ \left[x^{m}\right]\left(a\prod _{i\in J}\left(1+f_{i}\right)\right) =\left[x^{m}\right]a \qquad \text{for each }m\in \left\{ 0,1,\ldots ,n\right\} . \]

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theorem PowerSeries.coeff_mul_prod_one_add_eq {R : Type u_1} [CommRing R] {I : Type u_2} {a : PowerSeries R} {J : Finset I} {f : I → PowerSeries R} {n : ℕ} (hf : ∀ i ∈ J, ∀ m ≤ n, (coeff m) (f i) = 0) (m : ℕ) (hm : m ≤ n) :

(coeff m) (a * ∏ i ∈ J, (1 + f i)) = (coeff m) a

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Theorem 1.350

Let $\left(f_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a (possibly infinite) summable family of FPSs. Then, the family $\left(1+f_{i}\right)_{i\in I}$ is multipliable.

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theorem PowerSeries.multipliable_one_add_of_summable {R : Type u_1} [CommRing R] {I : Type u_2} {f : I → PowerSeries R} (hf : ∀ (n : ℕ), {i : I | (coeff n) (f i) ≠ 0}.Finite) :

Multipliable fun (i : I) => 1 + f i

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Proposition 1.351

If all but finitely many entries of a family $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ equal $1$ (that is, if all but finitely many $i\in I$ satisfy $\mathbf{a}_{i}=1$), then this family is multipliable.

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theorem PowerSeries.multipliable_of_finite_ne_one {R : Type u_1} [CommRing R] {I : Type u_2} (a : I → PowerSeries R) (h : {i : I | a i ≠ 1}.Finite) :

Multipliable a

$x^{n}$-approximators

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Definition 1.362

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a family of FPSs. Let $n\in \mathbb {N}$. An $x^{n}$-approximator for $\left(\mathbf{a}_{i}\right)_{i\in I}$ means a finite subset $M$ of $I$ that determines the first $n+1$ coefficients in the product of $\left(\mathbf{a}_{i}\right)_{i\in I}$. (In other words, $M$ has to determine the $x^{m}$-coefficient in the product of $\left(\mathbf{a}_{i}\right)_{i\in I}$ for each $m\in \left\{ 0,1,\ldots ,n\right\} $.)

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def PowerSeries.IsXnApproximator {R : Type u_1} [CommRing R] {I : Type u_2} (a : I → PowerSeries R) (M : Finset I) (n : ℕ) :

PowerSeries.IsXnApproximator a M n = ∀ m ≤ n, PowerSeries.DeterminesCoeffInProd a M m

docs source

abbrev AlgebraicCombinatorics.FPS.IsXnApproximator {K : Type u_1} [CommRing K] {ι : Type u_2} (a : ι → PowerSeries K) (n : ℕ) (U : Finset ι) :

AlgebraicCombinatorics.FPS.IsXnApproximator a n U = PowerSeries.IsXnApproximator a U n

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Lemma 1.363

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a multipliable family of FPSs. Let $n\in \mathbb {N}$. Then, there exists an $x^{n}$-approximator for $\left(\mathbf{a}_{i}\right)_{i\in I}$.

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theorem PowerSeries.exists_xn_approximator {R : Type u_1} [CommRing R] {I : Type u_2} [DecidableEq I] (a : I → PowerSeries R) (ha : Multipliable a) (n : ℕ) :

∃ (M : Finset I), IsXnApproximator a M n

docs source

theorem AlgebraicCombinatorics.FPS.multipliable_exists_approximator {K : Type u_1} [CommRing K] {ι : Type u_2} {a : ι → PowerSeries K} (h : Multipliable a) (n : ℕ) :

∃ (U : Finset ι), IsXnApproximator a n U

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Proposition 1.364

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a family of FPSs. Let $n\in \mathbb {N}$. Let $M$ be an $x^{n}$-approximator for $\left(\mathbf{a}_{i}\right)_{i\in I}$. Then:

(a) Every finite subset $J$ of $I$ satisfying $M\subseteq J\subseteq I$ satisfies

\[ \prod _{i\in J}\mathbf{a}_{i}\overset {x^{n}}{\equiv }\prod _{i\in M}\mathbf{a}_{i}. \]

(b) If the family $\left(\mathbf{a}_{i}\right)_{i\in I}$ is multipliable, then

\[ \prod _{i\in I}\mathbf{a}_{i}\overset {x^{n}}{\equiv }\prod _{i\in M}\mathbf{a}_{i}. \]

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theorem PowerSeries.xn_approximator_superset_xnEquiv {R : Type u_1} [CommRing R] {I : Type u_2} {a : I → PowerSeries R} {M : Finset I} {n : ℕ} (hM : IsXnApproximator a M n) {J : Finset I} (hMJ : M ⊆ J) :

∏ i ∈ J, a i ≡[x^n] ∏ i ∈ M, a i

docs source

theorem PowerSeries.tprod_xnEquiv_approximator {R : Type u_1} [CommRing R] {I : Type u_2} {a : I → PowerSeries R} (ha : Multipliable a) {M : Finset I} {n : ℕ} (hM : IsXnApproximator a M n) :

tprod a ha ≡[x^n] ∏ i ∈ M, a i

Properties of infinite products

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Proposition 1.365

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a family of FPSs. Let $J$ be a subset of $I$. Assume that the subfamilies $\left(\mathbf{a}_{i}\right)_{i\in J}$ and $\left(\mathbf{a}_{i}\right)_{i\in I\setminus J}$ are multipliable. Then:

(a) The entire family $\left(\mathbf{a}_{i}\right)_{i\in I}$ is multipliable.

(b) We have

\[ \prod _{i\in I}\mathbf{a}_{i}=\left(\prod _{i\in J}\mathbf{a}_{i}\right) \cdot \left(\prod _{i\in I\setminus J}\mathbf{a}_{i}\right). \]

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theorem PowerSeries.multipliable_of_union {R : Type u_1} [CommRing R] {I : Type u_2} {a : I → PowerSeries R} {J : Set I} (hJ : Multipliable fun (i : ↑J) => a ↑i) (hIJ : Multipliable fun (i : ↑(Set.univ \ J)) => a ↑i) :

Multipliable a

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theorem PowerSeries.tprod_eq_tprod_mul_tprod {R : Type u_1} [CommRing R] {I : Type u_2} {a : I → PowerSeries R} {J : Set I} (ha : Multipliable a) (hJ : Multipliable fun (i : ↑J) => a ↑i) (hIJ : Multipliable fun (i : ↑(Set.univ \ J)) => a ↑i) :

tprod a ha = tprod (fun (i : ↑J) => a ↑i) hJ * tprod (fun (i : ↑(Set.univ \ J)) => a ↑i) hIJ

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Proposition 1.366

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ and $\left(\mathbf{b}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be two multipliable families of FPSs. Then:

(a) The family $\left(\mathbf{a}_{i}\mathbf{b}_{i}\right)_{i\in I}$ is multipliable.

(b) We have

\[ \prod _{i\in I}\left(\mathbf{a}_{i}\mathbf{b}_{i}\right)=\left(\prod _{i\in I} \mathbf{a}_{i}\right)\cdot \left(\prod _{i\in I}\mathbf{b}_{i}\right). \]

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theorem PowerSeries.multipliable_mul {R : Type u_1} [CommRing R] {I : Type u_2} {a b : I → PowerSeries R} (ha : Multipliable a) (hb : Multipliable b) :

Multipliable fun (i : I) => a i * b i

docs source

theorem PowerSeries.tprod_mul_eq_mul_tprod {R : Type u_1} [CommRing R] {I : Type u_2} {a b : I → PowerSeries R} (ha : Multipliable a) (hb : Multipliable b) (hab : Multipliable fun (i : I) => a i * b i := ⋯) :

tprod (fun (i : I) => a i * b i) hab = tprod a ha * tprod b hb

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Proposition 1.367

(a) The family $\left(\frac{\mathbf{a}_{i}}{\mathbf{b}_{i}}\right)_{i\in I}$ is multipliable.

(b) We have

\[ \prod _{i\in I}\frac{\mathbf{a}_{i}}{\mathbf{b}_{i}} =\frac{\prod _{i\in I}\mathbf{a}_{i}}{\prod _{i\in I}\mathbf{b}_{i}}. \]

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theorem PowerSeries.multipliable_div {R : Type u_1} [CommRing R] {I : Type u_2} {a b : I → PowerSeries R} (ha : Multipliable a) (hb : Multipliable b) (hb_inv : ∀ (i : I), IsUnit ((coeff 0) (b i))) :

Multipliable fun (i : I) => a i * Ring.inverse (b i)

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theorem PowerSeries.tprod_div_eq_div_tprod {R : Type u_1} [CommRing R] {I : Type u_2} {a b : I → PowerSeries R} (ha : Multipliable a) (hb : Multipliable b) (hb_inv : ∀ (i : I), IsUnit ((coeff 0) (b i))) (hab : Multipliable fun (i : I) => a i * Ring.inverse (b i) := ⋯) :

tprod (fun (i : I) => a i * Ring.inverse (b i)) hab = tprod a ha * Ring.inverse (tprod b hb)

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Proposition 1.369

Let $\left(\mathbf{a}_{i}\right)_{i\in I}\in K\left[\left[x\right]\right]^{I}$ be a multipliable family of invertible FPSs. Then, any subfamily of $\left(\mathbf{a}_{i}\right)_{i\in I}$ is multipliable.

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theorem PowerSeries.multipliable_subfamily {R : Type u_1} [CommRing R] {I : Type u_2} {a : I → PowerSeries R} (ha : Multipliable a) (ha_inv : ∀ (i : I), IsUnit ((coeff 0) (a i))) (J : Set I) :

Multipliable fun (i : ↑J) => a ↑i

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Proposition 1.370

Let $S$ and $T$ be two sets. Let $f:S\rightarrow T$ be a bijection. Let $\left(\mathbf{a}_{t}\right)_{t\in T}\in K\left[\left[x\right]\right]^{T}$ be a multipliable family of FPSs. Then,

\[ \prod _{t\in T}\mathbf{a}_{t}=\prod _{s\in S}\mathbf{a}_{f\left(s\right)} \]

(and, in particular, the product on the right hand side is well-defined, i.e., the family $\left(\mathbf{a}_{f\left(s\right)}\right)_{s\in S}$ is multipliable).

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theorem PowerSeries.multipliable_reindex {R : Type u_1} [CommRing R] {S : Type u_3} {T : Type u_4} {f : S → T} (hf : Function.Bijective f) {a : T → PowerSeries R} (ha : Multipliable a) :

Multipliable (a ∘ f)

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theorem PowerSeries.tprod_reindex {R : Type u_1} [CommRing R] {S : Type u_3} {T : Type u_4} {f : S → T} (hf : Function.Bijective f) {a : T → PowerSeries R} (ha : Multipliable a) (haf : Multipliable (a ∘ f) := ⋯) :

tprod (a ∘ f) haf = tprod a ha

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Proposition 1.372

Let $\left(\mathbf{a}_{s}\right)_{s\in S}\in K\left[\left[x\right]\right]^{S}$ be a multipliable family of FPSs. Let $W$ be a set. Let $f:S\rightarrow W$ be a map. Assume that for each $w\in W$, the family $\left(\mathbf{a}_{s}\right)_{s\in S,\; f(s)=w}$ is multipliable. Then,

\[ \prod _{s\in S}\mathbf{a}_{s}=\prod _{w\in W}\; \; \prod _{\substack {s\in S;\\ f(s)=w}}\mathbf{a}_{s}. \]

(In particular, the right hand side is well-defined – i.e., the family $\left(\prod _{\substack {s\in S;\\ f(s)=w}}\mathbf{a}_{s}\right)_{w\in W}$ is multipliable.)

Note: The Lean formalization of this proposition (without sorry) requires the additional assumption that all FPSs are invertible (i.e., $[x^0]\mathbf{a}_s$ is a unit for each $s$), which is used through Proposition 1.369 to guarantee multipliability of subfamilies. The versions without this invertibility hypothesis are formalized with sorry.

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theorem PowerSeries.multipliable_prod_fibers_inv {R : Type u_1} [CommRing R] {S : Type u_3} {W : Type u_4} {f : S → W} {a : S → PowerSeries R} (ha : Multipliable a) (ha_inv : ∀ (s : S), IsUnit ((coeff 0) (a s))) (ha_fibers : ∀ (w : W), Multipliable fun (s : { s : S // f s = w }) => a ↑s := ⋯) :

Multipliable fun (w : W) => tprod (fun (s : { s : S // f s = w }) => a ↑s) ⋯

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theorem PowerSeries.tprod_eq_tprod_fibers_inv {R : Type u_1} [CommRing R] {S : Type u_3} {W : Type u_4} {f : S → W} {a : S → PowerSeries R} (ha : Multipliable a) (ha_inv : ∀ (s : S), IsUnit ((coeff 0) (a s))) (ha_fibers : ∀ (w : W), Multipliable fun (s : { s : S // f s = w }) => a ↑s := ⋯) (ha_outer : Multipliable fun (w : W) => tprod (fun (s : { s : S // f s = w }) => a ↑s) ⋯ := ⋯) :

tprod a ha = tprod (fun (w : W) => tprod (fun (s : { s : S // f s = w }) => a ↑s) ⋯) ha_outer

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Proposition 1.373 Fubini rule for infinite products of FPSs

Let $I$ and $J$ be two sets. Let $\left(\mathbf{a}_{(i,j)}\right)_{(i,j)\in I\times J}\in K\left[\left[x\right]\right]^{I\times J}$ be a multipliable family of FPSs. Assume that for each $i\in I$, the family $\left(\mathbf{a}_{(i,j)}\right)_{j\in J}$ is multipliable. Assume that for each $j\in J$, the family $\left(\mathbf{a}_{(i,j)}\right)_{i\in I}$ is multipliable. Then,

\[ \prod _{i\in I}\; \; \prod _{j\in J}\mathbf{a}_{(i,j)} =\prod _{(i,j)\in I\times J}\mathbf{a}_{(i,j)} =\prod _{j\in J}\; \; \prod _{i\in I}\mathbf{a}_{(i,j)}. \]

(In particular, all the products appearing in this equality are well-defined.)

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theorem PowerSeries.tprod_fubini {R : Type u_1} [CommRing R] {I : Type u_3} {J : Type u_4} {a : I × J → PowerSeries R} (ha : Multipliable a) (ha_I : ∀ (i : I), Multipliable fun (j : J) => a (i, j)) (_ha_J : ∀ (j : J), Multipliable fun (i : I) => a (i, j)) (ha_outer_I : Multipliable fun (i : I) => tprod (fun (j : J) => a (i, j)) ⋯ := ⋯) :

tprod a ha = tprod (fun (i : I) => tprod (fun (j : J) => a (i, j)) ⋯) ha_outer_I

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theorem PowerSeries.tprod_fubini_J {R : Type u_1} [CommRing R] {I : Type u_3} {J : Type u_4} {a : I × J → PowerSeries R} (ha : Multipliable a) (ha_J : ∀ (j : J), Multipliable fun (i : I) => a (i, j)) (ha_outer_J : Multipliable fun (j : J) => tprod (fun (i : I) => a (i, j)) ⋯ := ⋯) :

tprod a ha = tprod (fun (j : J) => tprod (fun (i : I) => a (i, j)) ⋯) ha_outer_J

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theorem PowerSeries.tprod_fubini_full {R : Type u_1} [CommRing R] {I : Type u_3} {J : Type u_4} {a : I × J → PowerSeries R} (ha : Multipliable a) (ha_I : ∀ (i : I), Multipliable fun (j : J) => a (i, j)) (ha_J : ∀ (j : J), Multipliable fun (i : I) => a (i, j)) (ha_outer_I : Multipliable fun (i : I) => tprod (fun (j : J) => a (i, j)) ⋯ := ⋯) (ha_outer_J : Multipliable fun (j : J) => tprod (fun (i : I) => a (i, j)) ⋯ := ⋯) :

tprod (fun (i : I) => tprod (fun (j : J) => a (i, j)) ⋯) ha_outer_I = tprod a ha ∧ tprod a ha = tprod (fun (j : J) => tprod (fun (i : I) => a (i, j)) ⋯) ha_outer_J

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theorem PowerSeries.multipliable_tprod_fubini {R : Type u_1} [CommRing R] {I : Type u_3} {J : Type u_4} {a : I × J → PowerSeries R} (ha : Multipliable a) (ha_I : ∀ (i : I), Multipliable fun (j : J) => a (i, j)) :

Multipliable fun (i : I) => tprod (fun (j : J) => a (i, j)) ⋯

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theorem PowerSeries.multipliable_tprod_fubini_J {R : Type u_1} [CommRing R] {I : Type u_3} {J : Type u_4} {a : I × J → PowerSeries R} (ha : Multipliable a) (ha_J : ∀ (j : J), Multipliable fun (i : I) => a (i, j)) :

Multipliable fun (j : J) => tprod (fun (i : I) => a (i, j)) ⋯

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Proposition 1.374 Fubini rule for infinite products, invertible case

Let $I$ and $J$ be two sets. Let $\left(\mathbf{a}_{(i,j)}\right)_{(i,j)\in I\times J}\in K\left[\left[x\right]\right]^{I\times J}$ be a multipliable family of invertible FPSs. Then,

\[ \prod _{i\in I}\; \; \prod _{j\in J}\mathbf{a}_{(i,j)} =\prod _{(i,j)\in I\times J}\mathbf{a}_{(i,j)} =\prod _{j\in J}\; \; \prod _{i\in I}\mathbf{a}_{(i,j)}. \]

(In particular, all the products appearing in this equality are well-defined.)

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theorem PowerSeries.fubini_prod_invertible {R : Type u_1} [CommRing R] {I : Type u_3} {J : Type u_4} {a : I × J → PowerSeries R} (ha : Multipliable a) (ha_inv : ∀ (p : I × J), IsUnit ((coeff 0) (a p))) :

(∀ (i : I), Multipliable fun (j : J) => a (i, j)) ∧ ∀ (j : J), Multipliable fun (i : I) => a (i, j)

Product rules (generalized distributive laws)

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Proposition 1.375

Let $L$ be a commutative ring. For every $n\in \mathbb {N}$, let $\left[ n\right] $ denote the set $\left\{ 1,2,\ldots ,n\right\} $.

Let $n\in \mathbb {N}$. For every $i\in \left[ n\right] $, let $p_{i,1},p_{i,2},\ldots ,p_{i,m_{i}}$ be finitely many elements of $L$. Then,

\begin{equation} \prod _{i=1}^{n}\ \ \sum _{k=1}^{m_{i}}p_{i,k}=\sum _{\left( k_{1},k_{2},\ldots ,k_{n}\right) \in \left[ m_{1}\right] \times \left[ m_{2}\right] \times \cdots \times \left[ m_{n}\right] }\ \ \prod _{i=1}^{n}p_{i,k_{i}}. \label{eq.lem.fps.prodrule-fin-fin.eq}\end{equation}

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theorem prod_sum_eq_sum_prod_finset {L : Type u_2} [CommSemiring L] {n : ℕ} {m : Fin n → ℕ} (p : (i : Fin n) → Fin (m i) → L) :

∏ i : Fin n, ∑ k : Fin (m i), p i k = ∑ f : (i : Fin n) → Fin (m i), ∏ i : Fin n, p i (f i)

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Proposition 1.376

For every $n\in \mathbb {N}$, let $\left[ n\right] $ denote the set $\left\{ 1,2,\ldots ,n\right\} $.

Let $n\in \mathbb {N}$. For every $i\in \left[ n\right] $, let $\left( p_{i,k}\right) _{k\in S_{i}}$ be a summable family of elements of $K\left[ \left[ x\right] \right] $. Then,

\begin{equation} \prod _{i=1}^{n}\ \ \sum _{k\in S_{i}}p_{i,k}=\sum _{\left( k_{1},k_{2},\ldots ,k_{n}\right) \in S_{1}\times S_{2}\times \cdots \times S_{n}}\ \ \prod _{i=1}^{n}p_{i,k_{i}}. \label{eq.lem.fps.prodrule-fin-inf.eq}\end{equation}

In particular, the family $\left( \prod _{i=1}^{n}p_{i,k_{i}}\right) _{\left( k_{1},k_{2},\ldots ,k_{n}\right) \in S_{1}\times S_{2}\times \cdots \times S_{n}}$ is summable.

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theorem prod_tsum_eq_tsum_prod_finset {K : Type u_1} [CommRing K] [TopologicalSpace K] [IsTopologicalRing K] [T2Space K] [DiscreteTopology K] {n : ℕ} {S : Fin n → Type u_2} (p : (i : Fin n) → S i → PowerSeries K) (hp : ∀ (i : Fin n), Summable (p i)) :

∏ i : Fin n, ∑' (k : S i), p i k = ∑' (f : (i : Fin n) → S i), ∏ i : Fin n, p i (f i)

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Definition 1.380

(a) A sequence $\left( k_{1},k_{2},k_{3},\ldots \right) $ is said to be essentially finite if all but finitely many $i\in \left\{ 1,2,3,\ldots \right\} $ satisfy $k_{i}=0$.

(b) A family $\left( k_{i}\right) _{i\in I}$ is said to be essentially finite if all but finitely many $i\in I$ satisfy $k_{i}=0$.

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def EssentiallyFinite {I : Type u_2} {M : Type u_3} [Zero M] (f : I → M) :

EssentiallyFinite f = (Function.support f).Finite

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Proposition 1.381

Let $S_{1},S_{2},S_{3},\ldots $ be infinitely many sets that all contain the number $0$. Set

\[ \overline{S}=\left\{ \left( i,k\right) \ \mid \ i\in \left\{ 1,2,3,\ldots \right\} \text{ and }k\in S_{i}\text{ and }k\neq 0\right\} . \]

For any $i\in \left\{ 1,2,3,\ldots \right\} $ and any $k\in S_{i}$, let $p_{i,k}$ be an element of $K\left[ \left[ x\right] \right] $. Assume that

\begin{equation} p_{i,0}=1\ \ \ \ \ \ \ \ \ \ \text{for any }i\in \left\{ 1,2,3,\ldots \right\} . \label{eq.prop.fps.prodrule-inf-infN.pi0=1}\end{equation}

Assume further that the family $\left( p_{i,k}\right) _{\left( i,k\right) \in \overline{S}}$ is summable. Then, the product $\prod _{i=1}^{\infty }\ \ \sum _{k\in S_{i}}p_{i,k}$ is well-defined (i.e., the family $\left( p_{i,k}\right) _{k\in S_{i}}$ is summable for each $i\in \left\{ 1,2,3,\ldots \right\} $, and the family $\left( \sum _{k\in S_{i}}p_{i,k}\right) _{i\in \left\{ 1,2,3,\ldots \right\} }$ is multipliable), and we have

\begin{equation} \prod _{i=1}^{\infty }\ \ \sum _{k\in S_{i}}p_{i,k}=\sum _{\substack {\left( k_{1},k_{2},k_{3},\ldots \right) \in S_{1}\times S_{2}\times S_{3}\times \cdots \\ \text{is essentially finite}}}\ \ \prod _{i=1}^{\infty }p_{i,k_{i}}. \label{eq.prop.fps.prodrule-inf-infN.eq}\end{equation}

In particular, the family $\left( \prod _{i=1}^{\infty }p_{i,k_{i}}\right) _{\left( k_{1},k_{2},k_{3},\ldots \right) \in S_{1}\times S_{2}\times S_{3}\times \cdots \text{ is essentially finite}}$ is summable.

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theorem tprod_tsum_eq_tsum_prod_essentiallyFinite_nat {K : Type u_1} [CommRing K] [TopologicalSpace K] [IsTopologicalRing K] [T2Space K] [DiscreteTopology K] {S : ℕ+ → Type u_2} [(i : ℕ+) → Zero (S i)] [(i : ℕ+) → DecidableEq (S i)] (p : (i : ℕ+) → S i → PowerSeries K) (hp_zero : ∀ (i : ℕ+), p i 0 = 1) (hp_summable : Summable fun (ik : SBar S) => p ik.fst ↑ik.snd) :

∏' (i : ℕ+), ∑' (k : S i), p i k = ∑' (f : { f : (i : ℕ+) → S i // ∀ᶠ (i : ℕ+) in Filter.cofinite , f i = 0 }), ∏' (i : ℕ+), p i (↑f i)

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Proposition 1.382

Let $I$ be a set. For any $i\in I$, let $S_{i}$ be a set that contains the number $0$. Set

\[ \overline{S}=\left\{ \left( i,k\right) \ \mid \ i\in I\text{ and }k\in S_{i}\text{ and }k\neq 0\right\} . \]

For any $i\in I$ and any $k\in S_{i}$, let $p_{i,k}$ be an element of $K\left[ \left[ x\right] \right] $. Assume that

\begin{equation} p_{i,0}=1\ \ \ \ \ \ \ \ \ \ \text{for any }i\in I. \label{eq.prop.fps.prodrule-inf-inf.pi0=1}\end{equation}

Assume further that the family $\left( p_{i,k}\right) _{\left( i,k\right) \in \overline{S}}$ is summable. Then, the product $\prod _{i\in I}\ \ \sum _{k\in S_{i}}p_{i,k}$ is well-defined (i.e., the family $\left( p_{i,k}\right) _{k\in S_{i}}$ is summable for each $i\in I$, and the family $\left( \sum _{k\in S_{i}}p_{i,k}\right) _{i\in I}$ is multipliable), and we have

\begin{equation} \prod _{i\in I}\ \ \sum _{k\in S_{i}}p_{i,k}=\sum _{\substack {\left( k_{i}\right) _{i\in I}\in \prod _{i\in I}S_{i}\\ \text{is essentially finite}}}\ \ \prod _{i\in I}p_{i,k_{i}}. \label{eq.prop.fps.prodrule-inf-inf.eq}\end{equation}

In particular, the family $\left( \prod _{i\in I}p_{i,k_{i}}\right) _{\left( k_{i}\right) _{i\in I}\in \prod _{i\in I}S_{i}\text{ is essentially finite}}$ is summable.

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theorem tprod_tsum_eq_tsum_prod_essentiallyFinite {K : Type u_1} [CommRing K] [TopologicalSpace K] [IsTopologicalRing K] [T2Space K] [DiscreteTopology K] {I : Type u_2} {S : I → Type u_3} [(i : I) → Zero (S i)] [(i : I) → DecidableEq (S i)] (p : (i : I) → S i → PowerSeries K) (hp_zero : ∀ (i : I), p i 0 = 1) (hp_summable : Summable fun (ik : (i : I) × { k : S i // k ≠ 0 }) => p ik.fst ↑ik.snd) :

∏' (i : I), ∑' (k : S i), p i k = ∑' (f : { f : (i : I) → S i // ∀ᶠ (i : I) in Filter.cofinite , f i = 0 }), ∏' (i : I), p i (↑f i)

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Proposition 1.383

Let $N$ be a finite set. For any $i\in N$, let $\left( p_{i,k}\right) _{k\in S_{i}}$ be a summable family of elements of $K\left[ \left[ x\right] \right] $. Then,

\begin{equation} \prod _{i\in N}\ \ \sum _{k\in S_{i}}p_{i,k}=\sum _{\left( k_{i}\right) _{i\in N}\in \prod _{i\in N}S_{i}}\ \ \prod _{i\in N}p_{i,k_{i}}. \label{eq.lem.fps.prodrule-fin-infJ.eq}\end{equation}

In particular, the family $\left( \prod _{i\in N}p_{i,k_{i}}\right) _{\left( k_{i}\right) _{i\in N}\in \prod _{i\in N}S_{i}}$ is summable.

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theorem prod_tsum_eq_tsum_prod_finset'_discrete {K : Type u_1} [CommRing K] [TopologicalSpace K] [IsTopologicalRing K] [T2Space K] [DiscreteTopology K] {N : Type u_2} [Fintype N] [DecidableEq N] {S : N → Type u_3} (p : (i : N) → S i → PowerSeries K) (hp : ∀ (i : N), Summable (p i)) :

∏ i : N, ∑' (k : S i), p i k = ∑' (f : (i : N) → S i), ∏ i : N, p i (f i)

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Lemma 1.384

Let $a\in K\left[ \left[ x\right] \right] $ be an FPS. Let $\left( f_{i}\right) _{i\in J}\in K\left[ \left[ x\right] \right] ^{J}$ be a summable family of FPSs. Let $n\in \mathbb {N}$. Assume that each $i\in J$ satisfies

\begin{equation} \left[ x^{m}\right] \left( f_{i}\right) =0\ \ \ \ \ \ \ \ \ \ \text{for each }m\in \left\{ 0,1,\ldots ,n\right\} . \label{eq.lem.fps.prod.irlv.inf.ass}\end{equation}

Then,

\[ \left[ x^{m}\right] \left( a\prod _{i\in J}\left( 1+f_{i}\right) \right) =\left[ x^{m}\right] a\ \ \ \ \ \ \ \ \ \ \text{for each }m\in \left\{ 0,1,\ldots ,n\right\} . \]

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theorem coeff_mul_tprod_one_add_eq_coeff {K : Type u_1} [CommRing K] [TopologicalSpace K] [T2Space K] {J : Type u_2} (a : PowerSeries K) (f : J → PowerSeries K) (_hf_summable : Summable f) (n : ℕ) (hf_order : ∀ (i : J), ∀ m ≤ n, (PowerSeries.coeff m) (f i) = 0) (m : ℕ) :

m ≤ n → (PowerSeries.coeff m) (a * ∏' (i : J), (1 + f i)) = (PowerSeries.coeff m) a

Another example

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Proposition 1.395 Euler

We have

\[ \prod _{i{\gt}0}\left( 1-x^{2i-1}\right) ^{-1}=\prod _{k{\gt}0}\left( 1+x^{k}\right) . \]

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theorem euler_odd_parts_identity {K : Type u_1} [CommRing K] [TopologicalSpace K] [IsTopologicalRing K] [T2Space K] [NoZeroDivisors K] :

∏' (i : ℕ+), Ring.inverse (1 - PowerSeries.X ^ (2 * ↑i - 1)) = ∏' (k : ℕ+), (1 + PowerSeries.X ^ ↑k)

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Theorem 1.396 Euler

Let $n\in \mathbb {N}$. Then, $d_{n}=o_{n}$, where

$d_{n}$ is the number of ways to write $n$ as a sum of distinct positive integers, without regard to the order;
$o_{n}$ is the number of ways to write $n$ as a sum of (not necessarily distinct) odd positive integers, without regard to the order.

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theorem euler_distinct_odd_partitions (n : ℕ) :

numDistinctPartitions n = numOddPartitions n

Infinite products and substitution

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Proposition 1.399

If $\left( f_{i}\right) _{i\in I}\in K\left[ \left[ x\right] \right] ^{I}$ is a multipliable family of FPSs, and if $g\in K\left[ \left[ x\right] \right] $ is an FPS satisfying $\left[ x^{0}\right] g=0$, then the family $\left( f_{i}\circ g\right) _{i\in I}\in K\left[ \left[ x\right] \right] ^{I}$ is multipliable as well and we have $\left( \prod _{i\in I}f_{i}\right) \circ g=\prod _{i\in I}\left( f_{i}\circ g\right) $.

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theorem tprod_rescale_substitution {K : Type u_1} [CommRing K] [TopologicalSpace K] [IsTopologicalRing K] [T2Space K] {I : Type u_2} (f : I → PowerSeries K) (a : K) (hf : Multipliable f) :

(PowerSeries.rescale a) (∏' (i : I), f i) = ∏' (i : I), (PowerSeries.rescale a) (f i)

Exponentials, logarithms and infinite products

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Proposition 1.400

Assume that $K$ is a commutative $\mathbb {Q}$-algebra. Let $\left( f_{i}\right) _{i\in I}\in K\left[ \left[ x\right] \right] ^{I}$ be a summable family of FPSs in $K\left[ \left[ x\right] \right] _{0}$. Then, $\left( \operatorname {Exp}f_{i}\right) _{i\in I}$ is a multipliable family of FPS in $K\left[ \left[ x\right] \right] _{1}$, and we have $\sum _{i\in I}f_{i}\in K\left[ \left[ x\right] \right] _{0}$ and

\[ \operatorname {Exp}\left( \sum _{i\in I}f_{i}\right) =\prod _{i\in I}\left( \operatorname {Exp}f_{i}\right) . \]

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theorem PowerSeries.Exp_sum {K : Type u_2} [CommRing K] [Algebra ℚ K] {I : Type u_3} (g : I → ↑PowerSeries₀) (hg : SummableFPS₀ g) (hg_mul : MultipliableFPS₁ fun (i : I) => Exp (g i) := ⋯) :

Exp (summableFPS₀Sum g hg) = multipliableFPS₁Prod (fun (i : I) => Exp (g i)) hg_mul

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theorem PowerSeries.Exp_multipliable_of_summable {K : Type u_2} [CommRing K] [Algebra ℚ K] {I : Type u_3} (g : I → ↑PowerSeries₀) (hg : SummableFPS₀ g) :

MultipliableFPS₁ fun (i : I) => Exp (g i)

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Proposition 1.401

Assume that $K$ is a commutative $\mathbb {Q}$-algebra. Let $\left( f_{i}\right) _{i\in I}\in K\left[ \left[ x\right] \right] ^{I}$ be a multipliable family of FPSs in $K\left[ \left[ x\right] \right] _{1}$. Then, $\left( \operatorname {Log}f_{i}\right) _{i\in I}$ is a summable family of FPS in $K\left[ \left[ x\right] \right] _{0}$, and we have $\prod \limits _{i\in I}f_{i}\in K\left[ \left[ x\right] \right] _{1}$ and

\[ \operatorname {Log}\left( \prod \limits _{i\in I}f_{i}\right) =\sum _{i\in I}\left( \operatorname {Log}f_{i}\right) . \]

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theorem PowerSeries.Log_tprod {K : Type u_2} [CommRing K] [Algebra ℚ K] {I : Type u_3} (f : I → ↑PowerSeries₁) (hf : MultipliableFPS₁ f) (hf_sum : SummableFPS₀ fun (i : I) => Log (f i) := ⋯) :

Log (multipliableFPS₁Prod f hf) = summableFPS₀Sum (fun (i : I) => Log (f i)) hf_sum

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theorem PowerSeries.Log_summable_of_multipliable {K : Type u_2} [CommRing K] [Algebra ℚ K] {I : Type u_3} (f : I → ↑PowerSeries₁) (hf : MultipliableFPS₁ f) :

SummableFPS₀ fun (i : I) => Log (f i)

The generating function of a weighted set

The theory

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Definition 1.402

(a) A weighted set is a set $A$ equipped with a function $w:A\rightarrow \mathbb {N}$, which is called the weight function of this weighted set. For each $a\in A$, the value $w\left( a\right) $ is denoted $\left\vert a\right\vert $ and is called the weight of $a$ (in our weighted set).

Usually, instead of explicitly specifying the weight function $w$ as a function, we will simply specify the weight $\left\vert a\right\vert $ for each $a\in A$. The weighted set consisting of the set $A$ and the weight function $w$ will be called $\left( A,w\right) $ or simply $A$ when the weight function $w$ is clear from the context.

(b) A weighted set $A$ is said to be finite-type if for each $n\in \mathbb {N}$, there are only finitely many $a\in A$ having weight $\left\vert a\right\vert =n$.

(c) If $A$ is a finite-type weighted set, then the weight generating function of $A$ is defined to be the FPS

\[ \sum _{a\in A}x^{\left\vert a\right\vert }=\sum _{n\in \mathbb {N}}\left( \text{\# of }a\in A\text{ having weight }n\right) \cdot x^{n}\in \mathbb {Z}\left[ \left[ x\right] \right] . \]

This FPS is denoted by $\overline{A}$.

(d) An isomorphism between two weighted sets $A$ and $B$ means a bijection $\rho :A\rightarrow B$ that preserves the weight (i.e., each $a\in A$ satisfies $\left\vert \rho \left( a\right) \right\vert =\left\vert a\right\vert $).

(e) We say that two weighted sets $A$ and $B$ are isomorphic (this is written $A\cong B$) if there exists an isomorphism between $A$ and $B$.

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structure WeightedSet (α : Type u_2) :

Type u_2

weight : α → ℕ
The weight function assigning a natural number to each element

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def WeightedSet.IsFiniteType {α : Type u_2} (W : WeightedSet α) :

W.IsFiniteType = ∀ (n : ℕ), {a : α | W.weight a = n}.Finite

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def WeightedSet.weightGenFun {R : Type u_1} [CommSemiring R] {α : Type u_2} (W : WeightedSet α) (hft : W.IsFiniteType) :

PowerSeries R

W.weightGenFun hft = PowerSeries.mk fun (n : ℕ) => ↑(W.countOfWeight hft n)

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structure WeightedSet.Isomorphism {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) :

Type (max u_2 u_3)

toEquiv : α ≃ β
The underlying bijection
weight_eq (a : α) : W₂.weight (self.toEquiv a) = W₁.weight a
The bijection preserves weights

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def WeightedSet.AreIsomorphic {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) :

(W₁ ≅ᵥ W₂) = Nonempty (W₁.Isomorphism W₂)

blueprint

Proposition 1.403

Let $A$ and $B$ be two isomorphic finite-type weighted sets. Then, $\overline{A}=\overline{B}$.

docs source

theorem WeightedSet.weightGenFun_eq_of_isomorphic {R : Type u_1} [CommSemiring R] {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) (hft₁ : W₁.IsFiniteType) (hft₂ : W₂.IsFiniteType) (h : W₁ ≅ᵥ W₂) :

W₁.weightGenFun hft₁ = W₂.weightGenFun hft₂

blueprint

Definition 1.404

Let $A$ and $B$ be two weighted sets. Then, the weighted set $A+B$ is defined to be the disjoint union of $A$ and $B$, with the weight function inherited from $A$ and $B$ (meaning that each element of $A$ has the same weight that it had in $A$, and each element of $B$ has the same weight that it had in $B$). Formally speaking, this means that $A+B$ is the set $\left( \left\{ 0\right\} \times A\right) \cup \left( \left\{ 1\right\} \times B\right) $, with the weight function given by

\begin{equation} \left\vert \left( 0,a\right) \right\vert =\left\vert a\right\vert \ \ \ \ \ \ \ \ \ \ \text{for each }a\in A \label{eq.def.gf-ws.djun.wt0}\end{equation}

and

\begin{equation} \left\vert \left( 1,b\right) \right\vert =\left\vert b\right\vert \ \ \ \ \ \ \ \ \ \ \text{for each }b\in B. \label{eq.def.gf-ws.djun.wt1}\end{equation}

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def WeightedSet.disjointUnion {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) :

WeightedSet (α ⊕ β)

W₁ +ᵥ W₂ = { weight := Sum.elim W₁.weight W₂.weight }

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Proposition 1.405

Let $A$ and $B$ be two finite-type weighted sets. Then, $A+B$ is finite-type, too, and satisfies $\overline{A+B}=\overline{A}+\overline{B}$.

docs source

theorem WeightedSet.disjointUnion_isFiniteType {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) (hft₁ : W₁.IsFiniteType) (hft₂ : W₂.IsFiniteType) :

(W₁ +ᵥ W₂).IsFiniteType

docs source

theorem WeightedSet.weightGenFun_disjointUnion {R : Type u_1} [CommSemiring R] {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) (hft₁ : W₁.IsFiniteType) (hft₂ : W₂.IsFiniteType) :

(W₁ +ᵥ W₂).weightGenFun ⋯ = W₁.weightGenFun hft₁ + W₂.weightGenFun hft₂

blueprint

Definition 1.406

Let $A$ and $B$ be two weighted sets. Then, the weighted set $A\times B$ is defined to be the Cartesian product of the sets $A$ and $B$ (that is, the set $\left\{ \left( a,b\right) \ \mid \ a\in A\text{ and }b\in B\right\} $), with the weight function defined as follows: For any $\left( a,b\right) \in A\times B$, we set

\begin{equation} \left\vert \left( a,b\right) \right\vert =\left\vert a\right\vert +\left\vert b\right\vert . \label{eq.def.gf-ws.prod.wt}\end{equation}

docs source

def WeightedSet.prod {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) :

WeightedSet (α × β)

W₁ ×ᵥ W₂ = { weight := fun (x : α × β) => match x with | (a, b) => W₁.weight a + W₂.weight b }

blueprint

Definition 1.408

Let $A$ be a weighted set. Then, $A^{k}$ (for $k\in \mathbb {N}$) means the weighted set $\underbrace{A\times A\times \cdots \times A}_{k\text{ times}}$.

docs source

theorem WeightedSet.prod_isFiniteType {α : Type u_2} {β : Type u_3} (W₁ : WeightedSet α) (W₂ : WeightedSet β) (hft₁ : W₁.IsFiniteType) (hft₂ : W₂.IsFiniteType) :

(W₁ ×ᵥ W₂).IsFiniteType

docs source

theorem WeightedSet.weightGenFun_prod {R : Type u_1} [CommSemiring R] {α : Type u_2} {β : Type u_3} [DecidableEq α] [DecidableEq β] (W₁ : WeightedSet α) (W₂ : WeightedSet β) (hft₁ : W₁.IsFiniteType) (hft₂ : W₂.IsFiniteType) :

(W₁ ×ᵥ W₂).weightGenFun ⋯ = W₁.weightGenFun hft₁ * W₂.weightGenFun hft₂

blueprint

Proposition 1.409

Let $A$ be a finite-type weighted set. Let $k\in \mathbb {N}$. Then, $A^{k}$ is finite-type, too, and satisfies $\overline{A^{k}}=\overline{A}^{k}$.

docs source

theorem WeightedSet.pow_isFiniteType {α : Type u_2} (W : WeightedSet α) (hft : W.IsFiniteType) (k : ℕ) :

(W.pow k).IsFiniteType

docs source

theorem WeightedSet.weightGenFun_pow {R : Type u_1} [CommSemiring R] {α : Type u_2} (W : WeightedSet α) (hft : W.IsFiniteType) (k : ℕ) :

(W.pow k).weightGenFun ⋯ = W.weightGenFun hft ^ k

Domino tilings

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Definition 1.412

(a) A shape means a subset of $\mathbb {Z}^{2}$.

We draw each $\left( i,j\right) \in \mathbb {Z}^{2}$ as a unit square with center at the point $\left( i,j\right) $ (in Cartesian coordinates); thus, a shape can be drawn as a cluster of squares.

(b) For any $n,m\in \mathbb {N}$, the shape $R_{n,m}$ (called the $n\times m$-rectangle) is defined to be

\[ \left\{ 1,2,\ldots ,n\right\} \times \left\{ 1,2,\ldots ,m\right\} =\left\{ \left( i,j\right) \in \mathbb {Z}^{2}\ \mid \ 1\leq i\leq n\text{ and }1\leq j\leq m\right\} . \]

(c) A domino means a size-$2$ shape of the form

\begin{align*} & \left\{ \left( i,j\right) ,\ \left( i+1,j\right) \right\} \text{ (a ``\emph{horizontal domino}'')}\ \ \ \ \ \ \ \ \ \ \text{or}\\ & \left\{ \left( i,j\right) ,\ \left( i,j+1\right) \right\} \text{ (a ``\emph{vertical domino}'')}\end{align*}

for some $\left( i,j\right) \in \mathbb {Z}^{2}$.

(d) A domino tiling of a shape $S$ is a set partition of $S$ into dominos (i.e., a set of disjoint dominos whose union is $S$).

(e) For any $n,m\in \mathbb {N}$, let $d_{n,m}$ be the # of domino tilings of $R_{n,m}$.

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abbrev DominoTilingsZ.Shape :

DominoTilingsZ.Shape = Set (ℤ × ℤ)

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def DominoTilingsZ.Rectangle (n m : ℕ) :

Shape

DominoTilingsZ.Rectangle n m = {p : ℤ × ℤ | 1 ≤ p.1 ∧ p.1 ≤ ↑n ∧ 1 ≤ p.2 ∧ p.2 ≤ ↑m}

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inductive DominoTilingsZ.Domino :

docs source

structure DominoTilingsZ.Tiling (S : Shape) :

dominos : Set Domino
The set of dominos in the tiling
pairwise_disjoint : self.dominos.PairwiseDisjoint Domino.toShape
The dominos are pairwise disjoint
cover : ⋃ d ∈ self.dominos, d.toShape = S
The union of dominos equals the shape

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def DominoTilingsZ.numTilings (n m : ℕ) :

TilingsHeight2.Isomorphism FaultfreeTilingsHeight2.tuples

d_[n,m] = Nat.card (DominoTilingsZ.Tiling (DominoTilingsZ.Rectangle n m))

blueprint

Lemma 1.413

Any domino tiling of a height-$2$ rectangle can be decomposed uniquely into a tuple of faultfree tilings of (usually smaller) height-$2$ rectangles, by cutting it along its faults. (If the original tiling was faultfree, then it decomposes into a $1$-tuple. If the original tiling was empty, then it decomposes into a $0$-tuple.)

Moreover, the sum of the weights of the faultfree tilings in the tuple is the weight of the original tiling. (In other words, if a tiling $T$ decomposes into the tuple $\left( T_{1},T_{2},\ldots ,T_{k}\right) $, then $\left\vert T\right\vert =\left\vert T_{1}\right\vert +\left\vert T_{2}\right\vert +\cdots +\left\vert T_{k}\right\vert $.)

In the language of weighted sets, this yields an isomorphism

\[ D\cong F^{0}+F^{1}+F^{2}+F^{3}+\cdots , \]

and therefore, by the infinite analogue of Proposition 1.405,

\[ \overline{D} =\overline{F^{0}+F^{1}+F^{2}+F^{3}+\cdots }=\overline{F^{0}}+\overline{F^{1}}+\overline{F^{2}}+\overline{F^{3}}+\cdots . \]

By Proposition 1.409, this equals

\[ \overline{F}^{0}+\overline{F}^{1}+\overline{F}^{2}+\overline{F}^{3}+\cdots =\frac{1}{1-\overline{F}}. \]

docs source

def DominoTilingsZ.tiling_decomposition_isomorphism :

One or more equations did not get rendered due to their size.

Limits of FPSs

Stabilization of scalars

blueprint

Definition 1.444

Let $\left( a_{i}\right) _{i\in \mathbb {N}}=\left( a_{0},a_{1},a_{2},\ldots \right) \in K^{\mathbb {N}}$ be a sequence of elements of $K$. Let $a\in K$.

We say that the sequence $\left( a_{i}\right) _{i\in \mathbb {N}}$ stabilizes to $a$ if there exists some $N\in \mathbb {N}$ such that

\[ \text{all integers }i\geq N\text{ satisfy }a_{i}=a. \]

If $a_{i}$ stabilizes to $a$ as $i\rightarrow \infty $, then we write $\lim \limits _{i\rightarrow \infty }a_{i}=a$ and say that $a$ is the limit (or eventual value) of $\left( a_{i}\right) _{i\in \mathbb {N}}$. This is legitimate, since $a$ is uniquely determined by the sequence $\left( a_{i}\right) _{i\in \mathbb {N}}$.

docs source

def Seq.StabilizesTo {K : Type u_1} (a : ℕ → K) (lim : K) :

Seq.StabilizesTo a lim = ∃ (N : ℕ), ∀ i ≥ N, a i = lim

Coefficientwise stabilization of FPSs

blueprint

Definition 1.453

Let $\left( f_{i}\right) _{i\in \mathbb {N}}\in K\left[ \left[ x\right] \right] ^{\mathbb {N}}$ be a sequence of FPSs over $K$. Let $f\in K\left[ \left[ x\right] \right] $ be an FPS.

We say that $\left( f_{i}\right) _{i\in \mathbb {N}}$ coefficientwise stabilizes to $f$ if for each $n\in \mathbb {N}$,

\[ \text{the sequence }\left( \left[ x^{n}\right] f_{i}\right) _{i\in \mathbb {N}}\text{ stabilizes to }\left[ x^{n}\right] f. \]

If $f_{i}$ coefficientwise stabilizes to $f$ as $i\rightarrow \infty $, then we write $\lim \limits _{i\rightarrow \infty }f_{i}=f$ and say that $f$ is the limit of $\left( f_{i}\right) _{i\in \mathbb {N}}$. This is legitimate, because $f$ is uniquely determined by the sequence $\left( f_{i}\right) _{i\in \mathbb {N}}$.

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def PowerSeries.CoeffStabilizesTo {K : Type u_1} [CommRing K] (f : ℕ → PowerSeries K) (lim : PowerSeries K) :

PowerSeries.CoeffStabilizesTo f lim = ∀ (n : ℕ), Seq.StabilizesTo (fun (i : ℕ) => (PowerSeries.coeff n) (f i)) ((PowerSeries.coeff n) lim)

Some properties of limits

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Theorem 1.461

Let $\left( f_{i}\right) _{i\in \mathbb {N}}\in K\left[ \left[ x\right] \right] ^{\mathbb {N}}$ be a sequence of FPSs. Assume that for each $n\in \mathbb {N}$, there exists some $g_{n}\in K$ such that the sequence $\left( \left[ x^{n}\right] f_{i}\right) _{i\in \mathbb {N}}$ stabilizes to $g_{n}$. Then, the sequence $\left( f_{i}\right) _{i\in \mathbb {N}}$ coefficientwise stabilizes to $\sum _{n\in \mathbb {N}}g_{n}x^{n}$. (That is, $\lim \limits _{i\rightarrow \infty }f_{i}=\sum _{n\in \mathbb {N}}g_{n}x^{n}$.)

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theorem PowerSeries.coeffStabilizesTo_of_forall_coeff_stabilizes {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {g : ℕ → K} (h : ∀ (n : ℕ), Seq.StabilizesTo (fun (i : ℕ) => (coeff n) (f i)) (g n)) :

CoeffStabilizesTo f (mk g)

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Lemma 1.462

Assume that $\left( f_{i}\right) _{i\in \mathbb {N}}$ is a sequence of FPSs, and that $f$ is an FPS such that $\lim \limits _{i\rightarrow \infty }f_{i}=f$. Then, for each $n\in \mathbb {N}$, there exists some integer $N\in \mathbb {N}$ such that

\[ \text{all integers }i\geq N\text{ satisfy }f_{i}\overset {x^{n}}{\equiv }f. \]

docs source

theorem PowerSeries.exists_xnEquiv_of_coeffStabilizesTo {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {lim : PowerSeries K} (h : CoeffStabilizesTo f lim) (n : ℕ) :

∃ (N : ℕ), ∀ i ≥ N, f i ≡[x^n] lim

blueprint

Proposition 1.463

Assume that $\left( f_{i}\right) _{i\in \mathbb {N}}$ and $\left( g_{i}\right) _{i\in \mathbb {N}}$ are two sequences of FPSs, and that $f$ and $g$ are two FPSs such that

\[ \lim \limits _{i\rightarrow \infty }f_{i}=f\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \lim \limits _{i\rightarrow \infty }g_{i}=g. \]

Then,

\[ \lim \limits _{i\rightarrow \infty }\left( f_{i}+g_{i}\right) =f+g\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \lim \limits _{i\rightarrow \infty }\left( f_{i}g_{i}\right) =fg. \]

docs source

theorem PowerSeries.coeffStabilizesTo_add {K : Type u_1} [CommRing K] {f g : ℕ → PowerSeries K} {lf lg : PowerSeries K} (hf : CoeffStabilizesTo f lf) (hg : CoeffStabilizesTo g lg) :

CoeffStabilizesTo (fun (i : ℕ) => f i + g i) (lf + lg)

docs source

theorem PowerSeries.coeffStabilizesTo_mul {K : Type u_1} [CommRing K] {f g : ℕ → PowerSeries K} {lf lg : PowerSeries K} (hf : CoeffStabilizesTo f lf) (hg : CoeffStabilizesTo g lg) :

CoeffStabilizesTo (fun (i : ℕ) => f i * g i) (lf * lg)

blueprint

Corollary 1.466

Let $k\in \mathbb {N}$. For each $i\in \left\{ 1,2,\ldots ,k\right\} $, let $f_{i}$ be an FPS, and let $\left( f_{i,n}\right) _{n\in \mathbb {N}}$ be a sequence of FPSs such that

\[ \lim \limits _{n\rightarrow \infty }\left( f_{i,n}\right) =f_{i} \]

(note that it is $n$, not $i$, that goes to $\infty $ here!). Then,

\[ \lim \limits _{n\rightarrow \infty }\sum _{i=1}^{k}f_{i,n}=\sum _{i=1}^{k}f_{i}\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \lim \limits _{n\rightarrow \infty }\prod _{i=1}^{k}f_{i,n}=\prod _{i=1}^{k}f_{i}. \]

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theorem PowerSeries.coeffStabilizesTo_finset_sum {K : Type u_1} [CommRing K] {ι : Type u_2} (s : Finset ι) {f : ι → ℕ → PowerSeries K} {lf : ι → PowerSeries K} (h : ∀ i ∈ s, CoeffStabilizesTo (f i) (lf i)) :

CoeffStabilizesTo (fun (n : ℕ) => ∑ i ∈ s, f i n) (∑ i ∈ s, lf i)

docs source

theorem PowerSeries.coeffStabilizesTo_finset_prod {K : Type u_1} [CommRing K] {ι : Type u_2} (s : Finset ι) {f : ι → ℕ → PowerSeries K} {lf : ι → PowerSeries K} (h : ∀ i ∈ s, CoeffStabilizesTo (f i) (lf i)) :

CoeffStabilizesTo (fun (n : ℕ) => ∏ i ∈ s, f i n) (∏ i ∈ s, lf i)

blueprint

Proposition 1.468

Assume that $\left( f_{i}\right) _{i\in \mathbb {N}}$ and $\left( g_{i}\right) _{i\in \mathbb {N}}$ are two sequences of FPSs, and that $f$ and $g$ are two FPSs such that

\[ \lim \limits _{i\rightarrow \infty }f_{i}=f\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \lim \limits _{i\rightarrow \infty }g_{i}=g. \]

Assume that each FPS $g_{i}$ is invertible. Then, $g$ is also invertible, and we have

\[ \lim \limits _{i\rightarrow \infty }\frac{f_{i}}{g_{i}}=\frac{f}{g}. \]

docs source

theorem PowerSeries.coeffStabilizesTo_mul_inv {K : Type u_1} [CommRing K] {f g : ℕ → PowerSeries K} {lf lg : PowerSeries K} (hf : CoeffStabilizesTo f lf) (hg : CoeffStabilizesTo g lg) (hunit : ∀ (i : ℕ), IsUnit (constantCoeff (g i))) :

CoeffStabilizesTo (fun (i : ℕ) => f i * (g i).invOfUnit ⋯.unit) (lf * lg.invOfUnit ⋯.unit)

blueprint

Proposition 1.469

Assume that $\left( f_{i}\right) _{i\in \mathbb {N}}$ and $\left( g_{i}\right) _{i\in \mathbb {N}}$ are two sequences of FPSs, and that $f$ and $g$ are two FPSs such that

\[ \lim \limits _{i\rightarrow \infty }f_{i}=f\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \lim \limits _{i\rightarrow \infty }g_{i}=g. \]

Assume that $\left[ x^{0}\right] g_{i}=0$ for each $i\in \mathbb {N}$. Then, $\left[ x^{0}\right] g=0$ and

\[ \lim \limits _{i\rightarrow \infty }\left( f_{i}\circ g_{i}\right) =f\circ g. \]

docs source

theorem PowerSeries.coeffStabilizesTo_subst {K : Type u_1} [CommRing K] {f g : ℕ → PowerSeries K} {lf lg : PowerSeries K} (hf : CoeffStabilizesTo f lf) (hg : CoeffStabilizesTo g lg) (hconst : ∀ (i : ℕ), constantCoeff (g i) = 0) :

CoeffStabilizesTo (fun (i : ℕ) => subst (g i) (f i)) (subst lg lf)

blueprint

Proposition 1.470

Let $\left( f_{n}\right) _{n\in \mathbb {N}}$ be a sequence of FPSs, and let $f$ be an FPS such that

\[ \lim \limits _{i\rightarrow \infty }f_{i}=f. \]

Then,

\[ \lim \limits _{i\rightarrow \infty }f_{i}^{\prime }=f^{\prime }. \]

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theorem PowerSeries.coeffStabilizesTo_derivativeFun {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {lf : PowerSeries K} (hf : CoeffStabilizesTo f lf) :

CoeffStabilizesTo (fun (i : ℕ) => (f i).derivativeFun) lf.derivativeFun

blueprint

Theorem 1.475

Let $\left( f_{n}\right) _{n\in \mathbb {N}}$ be a summable sequence of FPSs. Then,

\[ \lim \limits _{i\rightarrow \infty }\sum _{n=0}^{i}f_{n}=\sum _{n\in \mathbb {N}}f_{n}. \]

In other words, the infinite sum $\sum _{n\in \mathbb {N}}f_{n}$ is the limit of the finite partial sums $\sum _{n=0}^{i}f_{n}$.

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theorem PowerSeries.coeffStabilizesTo_partial_sum {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} (hf : IsSummable f) :

CoeffStabilizesTo (fun (i : ℕ) => ∑ j ∈ Finset.range (i + 1), f j) (tsum' f hf)

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Theorem 1.476

Let $\left( f_{n}\right) _{n\in \mathbb {N}}$ be a multipliable sequence of FPSs. Then,

\[ \lim \limits _{i\rightarrow \infty }\prod _{n=0}^{i}f_{n}=\prod _{n\in \mathbb {N}}f_{n}. \]

In other words, the infinite product $\prod _{n\in \mathbb {N}}f_{n}$ is the limit of the finite partial products $\prod _{n=0}^{i}f_{n}$.

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theorem PowerSeries.coeffStabilizesTo_partial_prod {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} (hf : IsMultipliable f) :

CoeffStabilizesTo (fun (i : ℕ) => ∏ j ∈ Finset.range (i + 1), f j) (tprod' f hf)

blueprint

Corollary 1.477

Each FPS is a limit of a sequence of polynomials. Indeed, if $a=\sum _{n\in \mathbb {N}}a_{n}x^{n}$ (with $a_{n}\in K$), then

\[ a=\lim \limits _{i\rightarrow \infty }\sum _{n=0}^{i}a_{n}x^{n}. \]

More precisely, the sequence of truncations $(\operatorname {trunc}_{i+1}(a))_{i \in \mathbb {N}}$ coefficientwise stabilizes to $a$.

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theorem PowerSeries.coeffStabilizesTo_trunc {K : Type u_1} [CommRing K] (a : PowerSeries K) :

CoeffStabilizesTo (fun (i : ℕ) => ↑((trunc (i + 1)) a)) a

blueprint

Theorem 1.478

Let $\left( f_{0},f_{1},f_{2},\ldots \right) $ be a sequence of FPSs such that $\lim \limits _{i\rightarrow \infty }\sum _{n=0}^{i}f_{n}$ exists. Then, the family $\left( f_{n}\right) _{n\in \mathbb {N}}$ is summable, and satisfies

\[ \sum _{n\in \mathbb {N}}f_{n}=\lim \limits _{i\rightarrow \infty }\sum _{n=0}^{i}f_{n}. \]

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theorem PowerSeries.isSummable_of_coeffStabilizesTo_partial_sum {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {lim : PowerSeries K} (h : CoeffStabilizesTo (fun (i : ℕ) => ∑ j ∈ Finset.range (i + 1), f j) lim) :

IsSummable f

docs source

theorem PowerSeries.tsum'_eq_of_coeffStabilizesTo_partial_sum {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {lim : PowerSeries K} (h : CoeffStabilizesTo (fun (i : ℕ) => ∑ j ∈ Finset.range (i + 1), f j) lim) :

tsum' f ⋯ = lim

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Theorem 1.479

Let $\left( f_{0},f_{1},f_{2},\ldots \right) $ be a sequence of FPSs such that $\lim \limits _{i\rightarrow \infty }\prod _{n=0}^{i}f_{n}$ exists. Then, the family $\left( f_{n}\right) _{n\in \mathbb {N}}$ is multipliable, and satisfies

\[ \prod _{n\in \mathbb {N}}f_{n}=\lim \limits _{i\rightarrow \infty }\prod _{n=0}^{i}f_{n}. \]

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theorem PowerSeries.isMultipliable_of_coeffStabilizesTo_partial_prod {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {lim : PowerSeries K} (h : CoeffStabilizesTo (fun (i : ℕ) => ∏ j ∈ Finset.range (i + 1), f j) lim) (hconst : ∀ (i : ℕ), constantCoeff (f i) = 1) :

IsMultipliable f

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theorem PowerSeries.tprod'_eq_of_coeffStabilizesTo_partial_prod {K : Type u_1} [CommRing K] {f : ℕ → PowerSeries K} {lim : PowerSeries K} (h : CoeffStabilizesTo (fun (i : ℕ) => ∏ j ∈ Finset.range (i + 1), f j) lim) (hconst : ∀ (i : ℕ), constantCoeff (f i) = 1) :

tprod' f ⋯ = lim

Laurent power series

Motivation

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Theorem 1.481

Each $n\in \mathbb {N}$ has a unique binary representation.

docs source

theorem AlgebraicCombinatorics.FPS.Laurent.binaryRepresentation_exists_unique (n : ℕ) :

∃! s : Finset ℕ, BinaryRepresentation n s

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theorem AlgebraicCombinatorics.binaryRepresentation_unique (n : ℕ) (r₁ r₂ : BinaryRepresentation n) :

r₁.bits = r₂.bits

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theorem AlgebraicCombinatorics.binaryRepresentation_exists (n : ℕ) :

Nonempty (BinaryRepresentation n)

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Theorem 1.483

Each integer $n$ has a unique balanced ternary representation.

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theorem AlgebraicCombinatorics.FPS.Laurent.balancedTernary_exists_unique (n : ℤ) :

∃! f : ℕ →₀ ℤ , (∀ (i : ℕ), f i ∈ btDigits) ∧ balancedTernaryValue f = n

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theorem AlgebraicCombinatorics.balancedTernaryRepresentation_unique (n : ℤ) (r₁ r₂ : BalancedTernaryRepresentation n) :

r₁.digits = r₂.digits

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theorem AlgebraicCombinatorics.balancedTernaryRepresentation_exists (n : ℤ) :

Nonempty (BalancedTernaryRepresentation n)

The space $K\left[ \left[ x^{\pm}\right] \right] $

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Definition 1.494

Let $K\left[ \left[ x^{\pm }\right] \right] $ be the $K$-module $K^{\mathbb {Z}}$ of all families $\left( a_{n}\right) _{n\in \mathbb {Z}}=\left( \ldots ,a_{-2},a_{-1},a_{0},a_{1},a_{2},\ldots \right) $ of elements of $K$. Its addition and its scaling are defined entrywise:

\begin{align*} \left( a_{n}\right) _{n\in \mathbb {Z}}+\left( b_{n}\right) _{n\in \mathbb {Z}} & =\left( a_{n}+b_{n}\right) _{n\in \mathbb {Z}};\\ \lambda \left( a_{n}\right) _{n\in \mathbb {Z}} & =\left( \lambda a_{n}\right) _{n\in \mathbb {Z}}\ \ \ \ \ \ \ \ \ \ \text{for each }\lambda \in K. \end{align*}

An element of $K\left[ \left[ x^{\pm }\right] \right] $ will be called a doubly infinite power series. We use the notation $\sum _{n\in \mathbb {Z}}a_{n}x^{n}$ for a family $\left( a_{n}\right) _{n\in \mathbb {Z}}\in K\left[ \left[ x^{\pm }\right] \right] $.

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def AlgebraicCombinatorics.DoublyInfinitePowerSeries (K : Type u_1) [Semiring K] :

Type u_1

AlgebraicCombinatorics.DoublyInfinitePowerSeries K = (ℤ → K)

Laurent polynomials

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Definition 1.497

Let $K\left[ x^{\pm }\right] $ be the $K$-submodule of $K\left[ \left[ x^{\pm }\right] \right] $ consisting of all essentially finite families $\left( a_{n}\right) _{n\in \mathbb {Z}}$. The elements of $K\left[ x^{\pm }\right] $ are called Laurent polynomials in the indeterminate $x$ over $K$.

We define a multiplication on $K\left[ x^{\pm }\right] $ by setting

\[ \left( a_{n}\right) _{n\in \mathbb {Z}}\cdot \left( b_{n}\right) _{n\in \mathbb {Z}}=\left( c_{n}\right) _{n\in \mathbb {Z}},\ \ \ \ \ \ \ \ \ \ \text{where}\ \ \ \ \ \ \ \ \ \ c_{n}=\sum _{i\in \mathbb {Z}}a_{i}b_{n-i}. \]

The sum $\sum _{i\in \mathbb {Z}}a_{i}b_{n-i}$ is well-defined because it is essentially finite.

We define an element $x\in K\left[ x^{\pm }\right] $ by

\[ x=\left( \delta _{i,1}\right) _{i\in \mathbb {Z}}. \]

In Mathlib, Laurent polynomials are represented as finitely supported functions $\mathbb {Z} \to K$ (the group algebra of $\mathbb {Z}$ over $K$).

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abbrev AlgebraicCombinatorics.FPS.Laurent.LaurentPoly (K : Type u_2) [CommRing K] :

Type u_2

AlgebraicCombinatorics.FPS.Laurent.LaurentPoly K = LaurentPolynomial K

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theorem AlgebraicCombinatorics.FPS.Laurent.laurentPoly_mul_coeff {K : Type u_1} [CommRing K] (a b : LaurentPoly K) (n : ℤ) :

(a * b) n = ∑ i ∈ a.support, a i * b (n - i)

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def AlgebraicCombinatorics.FPS.Laurent.laurentPoly_X {K : Type u_1} [CommRing K] :

LaurentPoly K

AlgebraicCombinatorics.FPS.Laurent.laurentPoly_X = LaurentPolynomial.T 1

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theorem AlgebraicCombinatorics.laurentPolynomial_one_eq_T_zero (K : Type u_1) [CommSemiring K] :

1 = LaurentPolynomial.T 0

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Theorem 1.498

The $K$-module $K\left[ x^{\pm }\right] $, equipped with the multiplication defined above, is a commutative $K$-algebra. Its unity is $\left( \delta _{i,0}\right) _{i\in \mathbb {Z}}$. The element $x$ is invertible in this $K$-algebra.

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def AlgebraicCombinatorics.FPS.Laurent.laurentPolynomial_commRing {K : Type u_1} [CommRing K] :

CommRing (LaurentPolynomial K)

AlgebraicCombinatorics.FPS.Laurent.laurentPolynomial_commRing = inferInstance

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def AlgebraicCombinatorics.FPS.Laurent.laurentPolynomial_algebra {K : Type u_1} [CommRing K] :

Algebra K (LaurentPolynomial K)

AlgebraicCombinatorics.FPS.Laurent.laurentPolynomial_algebra = inferInstance

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theorem AlgebraicCombinatorics.FPS.Laurent.T_isUnit {K : Type u_1} [CommRing K] :

IsUnit (LaurentPolynomial.T 1)

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instance AlgebraicCombinatorics.laurentPolynomial_commSemiring (K : Type u_1) [CommSemiring K] :

CommSemiring (LaurentPolynomial K)

AlgebraicCombinatorics.laurentPolynomial_commSemiring K = inferInstance

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instance AlgebraicCombinatorics.laurentPolynomial_algebra (K : Type u_1) [CommSemiring K] :

Algebra K (LaurentPolynomial K)

AlgebraicCombinatorics.laurentPolynomial_algebra K = inferInstance

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theorem AlgebraicCombinatorics.laurentPolynomial_T_isUnit (K : Type u_1) [CommSemiring K] :

IsUnit (LaurentPolynomial.T 1)

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Proposition 1.505

Any doubly infinite power series $a=\left( a_{i}\right) _{i\in \mathbb {Z}}\in K\left[ \left[ x^{\pm }\right] \right] $ satisfies

\[ a=\sum _{i\in \mathbb {Z}}a_{i}x^{i}. \]

Here, the powers $x^{i}$ are taken in the Laurent polynomial ring $K\left[ x^{\pm }\right] $, but the infinite sum $\sum _{i\in \mathbb {Z}}a_{i}x^{i}$ is taken in the $K$-module $K\left[ \left[ x^{\pm }\right] \right] $.

For Laurent polynomials, the sum is finite and the identity holds in $K[x^{\pm }]$ itself. For Laurent series, the identity is formalized using a summable family construction.

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theorem AlgebraicCombinatorics.FPS.Laurent.eq_sum_T {K : Type u_1} [CommRing K] (a : LaurentPolynomial K) :

a = ∑ i ∈ a.support, LaurentPolynomial.C (a i) * LaurentPolynomial.T i

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theorem AlgebraicCombinatorics.FPS.Laurent.laurentSeries_eq_hsum_single {K : Type u_1} [CommRing K] (x : LaurentSeries K) :

x = (singleFamily x).hsum

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theorem AlgebraicCombinatorics.laurentPolynomial_eq_sum (K : Type u_1) [CommSemiring K] (f : LaurentPolynomial K) :

f = Finsupp.sum f fun (n : ℤ) (a : K) => LaurentPolynomial.C a * LaurentPolynomial.T n

Laurent series

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Definition 1.506

We let $K\left( \left( x\right) \right) $ be the subset of $K\left[ \left[ x^{\pm }\right] \right] $ consisting of all families $\left( a_{i}\right) _{i\in \mathbb {Z}}\in K\left[ \left[ x^{\pm }\right] \right] $ such that the sequence $\left( a_{-1},a_{-2},a_{-3},\ldots \right) $ is essentially finite – i.e., such that all sufficiently low $i\in \mathbb {Z}$ satisfy $a_{i}=0$.

The elements of $K\left( \left( x\right) \right) $ are called Laurent series in one indeterminate $x$ over $K$.

In Mathlib, Laurent series are implemented as Hahn series over $\mathbb {Z}$ with coefficients in $K$, which are functions $\mathbb {Z} \to K$ whose support is well-founded (equivalently, bounded below). The formalization also provides a predicate on doubly infinite power series that captures this textbook definition.

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def AlgebraicCombinatorics.DoublyInfinitePowerSeries.IsLaurentSeries {K : Type u_1} [Semiring K] (f : DoublyInfinitePowerSeries K) :

f.IsLaurentSeries = ∃ (N : ℤ), ∀ n < N, f.coeff n = 0

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def AlgebraicCombinatorics.FPS.Laurent.laurentSeries_commRing {K : Type u_1} [CommRing K] :

CommRing (LaurentSeries K)

AlgebraicCombinatorics.FPS.Laurent.laurentSeries_commRing = inferInstance

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Theorem 1.509

The subset $K\left( \left( x\right) \right) $ is a $K$-submodule of $K\left[ \left[ x^{\pm }\right] \right] $. It has a multiplication given by the same convolution rule as $K\left[ x^{\pm }\right] $: namely,

The sum $\sum _{i\in \mathbb {Z}}a_{i}b_{n-i}$ is well-defined because for sufficiently low $i$ we have $a_i = 0$, and for sufficiently high $i$ we have $b_{n-i} = 0$.

Equipped with this multiplication, $K\left( \left( x\right) \right) $ is a commutative $K$-algebra with unity $\left( \delta _{i,0}\right) _{i\in \mathbb {Z}}$. The element $x$ is invertible in this algebra.

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def AlgebraicCombinatorics.FPS.Laurent.laurentSeries_commRing {K : Type u_1} [CommRing K] :

CommRing (LaurentSeries K)

AlgebraicCombinatorics.FPS.Laurent.laurentSeries_commRing = inferInstance

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def AlgebraicCombinatorics.FPS.Laurent.laurentSeries_algebra {K : Type u_1} [CommRing K] :

Algebra K (LaurentSeries K)

AlgebraicCombinatorics.FPS.Laurent.laurentSeries_algebra = inferInstance

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theorem AlgebraicCombinatorics.FPS.Laurent.LaurentSeries_X_isUnit {K : Type u_1} [CommRing K] :

IsUnit ((HahnSeries.single 1) 1)

Multivariate FPSs

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Proposition 1.525

Let $f_{0},f_{1},f_{2},\ldots $ and $g_{0},g_{1},g_{2},\ldots $ be FPSs in a single variable $x$ such that

\begin{equation} \sum _{k\in \mathbb {N}}f_{k}y^{k}=\sum _{k\in \mathbb {N}}g_{k}y^{k}\ \ \ \ \ \ \ \ \ \ \text{in }K\left[ \left[ x,y\right] \right] . \label{eq.prop.fps.mulvar.comp-y-coeff.ass} \end{equation}

Then, $f_{k}=g_{k}$ for each $k\in \mathbb {N}$.

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theorem AlgebraicCombinatorics.eq_of_embedUnivInBiv_eq {R : Type u_1} [CommSemiring R] (f g : ℕ → PowerSeries R) (h : embedUnivInBiv f = embedUnivInBiv g) :

f = g

Partition basics

Definitions

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Definition 2.1

(a) An (integer) partition means a (finite) weakly decreasing tuple of positive integers – i.e., a finite tuple $\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{m}\right) $ of positive integers such that $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{m}$.

Thus, partitions are the same as weakly decreasing compositions. Hence, the notions of size and length of a partition are automatically defined, since we have defined them for compositions (in Definition 1.291).

(b) The parts of a partition $\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{m}\right) $ are simply its entries $\lambda _{1},\lambda _{2},\ldots ,\lambda _{m}$.

(c) Let $n\in \mathbb {Z}$. A partition of $n$ means a partition whose size is $n$.

(d) Let $n\in \mathbb {Z}$ and $k\in \mathbb {N}$. A partition of $n$ into $k$ parts is a partition whose size is $n$ and whose length is $k$.

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structure Nat.Partition (n : ℕ) :

parts : Multiset ℕ
positive integers summing to n
parts_pos {i : ℕ} : i ∈ self.parts → 0 < i
proof that the parts are positive
parts_sum : self.parts.sum = n
proof that the parts sum to n

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theorem Nat.Partition.size_eq {n : ℕ} (p : n.Partition) :

p.parts.sum = n

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theorem Nat.Partition.parts_pos' {n : ℕ} (p : n.Partition) (i : ℕ) (hi : i ∈ p.parts) :

0 < i

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def Nat.Partition.ofList' {n : ℕ} (l : List ℕ) (hl_pos : ∀ i ∈ l, 0 < i) (hl_sum : l.sum = n) :

n.Partition

Nat.Partition.ofList' l hl_pos hl_sum = { parts := ↑l, parts_pos := ⋯, parts_sum := ⋯ }

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theorem Nat.Partition.eq_iff_parts_eq {n : ℕ} (p q : n.Partition) :

p = q ↔ p.parts = q.parts

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Definition 2.10

(a) Let $n\in \mathbb {Z}$ and $k\in \mathbb {N}$. Then, we set

\[ p_{k}\left( n\right) :=\left( \text{\# of partitions of }n\text{ into }k\text{ parts}\right) . \]

(b) Let $n\in \mathbb {Z}$. Then, we set

\[ p\left( n\right) :=\left( \text{\# of partitions of }n\right) . \]

This is called the $n$-th partition number.

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def Nat.Partition.partitionCount (n : ℕ) :

Nat.Partition.partitionCount n = Fintype.card n.Partition

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def Nat.Partition.partsCount (k n : ℕ) :

Nat.Partition.partsCount k n = {p : n.Partition | p.parts.card = k}.card

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def Nat.Partition.partsLeqCount (m n : ℕ) :

Nat.Partition.partsLeqCount m n = (Nat.Partition.restricted n fun (x : ℕ) => x ≤ m).card

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def Nat.Partition.partsInCount (I : Set ℕ) [DecidablePred fun (x : ℕ) => x ∈ I] (n : ℕ) :

Nat.Partition.partsInCount I n = (Nat.Partition.restricted n fun (x : ℕ) => x ∈ I).card

Simple properties of partition numbers

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Definition 2.11

We will use the Iverson bracket notation: If $\mathcal{A}$ is a logical statement, then $\left[ \mathcal{A} \right]$ means the truth value of $\mathcal{A}$; this is the integer $\begin{cases} 1, & \text{if }\mathcal{A}\text{ is true};\\ 0, & \text{if }\mathcal{A}\text{ is false}. \end{cases} $

Note that the Kronecker delta notation is a particular case of the Iverson bracket: We have

\[ \delta _{i,j}=\left[ i=j \right]\ \ \ \ \ \ \ \ \ \ \text{for any objects }i\text{ and }j. \]

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abbrev IversonBracket.iverson (P : Prop) [Decidable P] {α : Type u_1} [Zero α] [One α] :

IversonBracket.iverson P = if P then 1 else 0

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theorem IversonBracket.iverson_true {α : Type u_1} [Zero α] [One α] {P : Prop} [Decidable P] (h : P) :

iverson P = 1

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theorem IversonBracket.iverson_false {α : Type u_1} [Zero α] [One α] {P : Prop} [Decidable P] (h : ¬P) :

iverson P = 0

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theorem IversonBracket.kronecker_eq_iverson {α : Type u_1} [DecidableEq α] {R : Type u_2} [Zero R] [One R] (i j : α) :

(if i = j then 1 else 0) = iverson (i = j)

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Convention 2.17

We agree to say that the largest part of the empty partition $\left( {}\right) $ is $0$ (even though this partition has no parts).

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theorem FloorCeiling.floor_def (a : ℝ) (n : ℤ) :

n ≤ ⌊a⌋ ↔ ↑n ≤ a

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theorem FloorCeiling.ceil_def (a : ℝ) (n : ℤ) :

⌈a⌉ ≤ n ↔ a ≤ ↑n

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theorem FloorCeiling.floor_of_int (n : ℤ) :

⌊↑n⌋ = n

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theorem FloorCeiling.ceil_of_int (n : ℤ) :

⌈↑n⌉ = n

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theorem FloorCeiling.nat_div_eq_floor (n m : ℕ) (_hm : m ≠ 0) :

n / m = ⌊↑n / ↑m⌋₊

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Proposition 2.22

Let $n\in \mathbb {Z}$ and $k\in \mathbb {N}$.

(a) We have $p_{k}\left( n\right) =0$ whenever $n{\lt}0$ and $k\in \mathbb {N}$.

(b) We have $p_{k}\left( n\right) =0$ whenever $k{\gt}n$.

(c) We have $p_{0}\left( n\right) =\left[ n=0 \right]$.

(d) We have $p_{1}\left( n\right) =\left[ n{\gt}0 \right]$.

(e) We have $p_{k}\left( n\right) =p_{k}\left( n-k\right) +p_{k-1}\left( n-1\right) $ whenever $k{\gt}0$.

(f) We have $p_{2}\left( n\right) =\left\lfloor n/2\right\rfloor $ whenever $n\in \mathbb {N}$.

(g) We have $p\left( n\right) =p_{0}\left( n\right) +p_{1}\left( n\right) +\cdots +p_{n}\left( n\right) $ whenever $n\in \mathbb {N}$.

(h) We have $p\left( n\right) =0$ whenever $n{\lt}0$.

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theorem Nat.Partition.partsCount_of_gt {k n : ℕ} (h : k > n) :

partsCount k n = 0

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theorem Nat.Partition.partsCount_zero (n : ℕ) :

partsCount 0 n = if n = 0 then 1 else 0

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theorem Nat.Partition.partsCount_one (n : ℕ) :

partsCount 1 n = if n > 0 then 1 else 0

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theorem Nat.Partition.partsCount_recurrence {k n : ℕ} (hk : k > 0) (hn : n > 0) :

partsCount k n = partsCount k (n - k) + partsCount (k - 1) (n - 1)

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theorem Nat.Partition.partsCount_two (n : ℕ) :

partsCount 2 n = n / 2

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theorem Nat.Partition.partitionCount_sum (n : ℕ) :

partitionCount n = ∑ k ∈ Finset.range (n + 1), partsCount k n

The generating function

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Theorem 2.30

In the FPS ring $\mathbb {Z}\left[ \left[ x\right] \right] $, we have

\[ \sum _{n\in \mathbb {N}}p\left( n\right) x^{n}=\prod _{k=1}^{\infty }\frac{1}{1-x^{k}}. \]

(The product on the right hand side is well-defined, since multiplying a FPS by $\frac{1}{1-x^{k}}$ does not affect its first $k$ coefficients.)

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theorem Nat.Partition.partitionCount_genFun {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [T2Space R] [IsTopologicalSemiring R] :

HasProd (fun (k : ℕ) => ∑' (j : ℕ), PowerSeries.X ^ ((k + 1) * j)) (PowerSeries.mk fun (n : ℕ) => ↑(partitionCount n))

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Theorem 2.31

Let $m\in \mathbb {N}$. For each $n\in \mathbb {N}$, let $p_{\operatorname {parts}\leq m}\left( n\right) $ be the # of partitions $\lambda $ of $n$ such that all parts of $\lambda $ are $\leq m$. Then,

\[ \sum _{n\in \mathbb {N}}p_{\operatorname {parts}\leq m}\left( n\right) x^{n}=\prod _{k=1}^{m}\frac{1}{1-x^{k}}. \]

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theorem Nat.Partition.partitionCount_genFun_partsLeq {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [T2Space R] [IsTopologicalSemiring R] (m : ℕ) :

HasProd (fun (k : ℕ) => if k + 1 ≤ m then ∑' (j : ℕ), PowerSeries.X ^ ((k + 1) * j) else 1) (PowerSeries.mk fun (n : ℕ) => ↑(partsLeqCount m n))

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theorem Nat.Partition.partitionCount_genFun_partsLeq_finprod {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [T2Space R] [IsTopologicalSemiring R] (m : ℕ) :

∏ k ∈ Finset.range m, ∑' (j : ℕ), PowerSeries.X ^ ((k + 1) * j) = PowerSeries.mk fun (n : ℕ) => ↑(restricted n fun (x : ℕ) => x ≤ m).card

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Theorem 2.32

Let $I$ be a subset of $\left\{ 1,2,3,\ldots \right\} $. For each $n\in \mathbb {N}$, let $p_{I}\left( n\right) $ be the # of partitions $\lambda $ of $n$ such that all parts of $\lambda $ belong to $I$. Then,

\[ \sum _{n\in \mathbb {N}}p_{I}\left( n\right) x^{n}=\prod _{k\in I}\frac{1}{1-x^{k}}. \]

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theorem Nat.Partition.partitionCount_genFun_partsIn {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [T2Space R] [IsTopologicalSemiring R] (I : Set ℕ) [DecidablePred fun (x : ℕ) => x ∈ I] :

HasProd (fun (k : ℕ) => if k + 1 ∈ I then ∑' (j : ℕ), PowerSeries.X ^ ((k + 1) * j) else 1) (PowerSeries.mk fun (n : ℕ) => ↑(partsInCount I n))

Odd parts and distinct parts

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Definition 2.33

Let $n\in \mathbb {Z}$.

(a) A partition of $n$ into odd parts means a partition of $n$ whose all parts are odd.

(b) A partition of $n$ into distinct parts means a partition of $n$ whose parts are distinct.

(c) Let

\begin{align*} p_{\operatorname {odd}}\left( n\right) & :=\left( \text{\# of partitions of }n\text{ into odd parts}\right) \ \ \ \ \ \ \ \ \ \ \text{and}\\ p_{\operatorname {dist}}\left( n\right) & :=\left( \text{\# of partitions of }n\text{ into distinct parts}\right) . \end{align*}

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def Nat.Partition.IsOddParts {n : ℕ} (p : n.Partition) :

p.IsOddParts = ∀ i ∈ p.parts, Odd i

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def Nat.Partition.IsDistinctParts {n : ℕ} (p : n.Partition) :

p.IsDistinctParts = p.parts.Nodup

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def Nat.Partition.oddPartsCount (n : ℕ) :

Nat.Partition.oddPartsCount n = (Nat.Partition.odds n).card

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def Nat.Partition.distinctPartsCount (n : ℕ) :

Nat.Partition.distinctPartsCount n = (Nat.Partition.distincts n).card

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Theorem 2.37 Euler’s odd-distinct identity

We have $p_{\operatorname {odd}}\left( n\right) =p_{\operatorname {dist}}\left( n\right) $ for each $n\in \mathbb {N}$.

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theorem Nat.Partition.odd_eq_distinct (n : ℕ) :

(odds n).card = (distincts n).card

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theorem Nat.Partition.oddPartsCount_eq_distinctPartsCount (n : ℕ) :

oddPartsCount n = distinctPartsCount n

Partitions with a given largest part

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Proposition 2.46

Let $n\in \mathbb {N}$ and $k\in \mathbb {N}$. Then,

\[ p_{k}\left( n\right) =\left( \text{\# of partitions of }n\text{ whose largest part is }k\right) . \]

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theorem Nat.Partition.partsCount_eq_largestPartCount (n k : ℕ) :

partsCount k n = {p : n.Partition | p.largestPart = k}.card

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Corollary 2.47

Let $n\in \mathbb {N}$ and $k\in \mathbb {N}$. Then,

\begin{align*} & p_{0}\left( n\right) +p_{1}\left( n\right) +\cdots +p_{k}\left( n\right) \\ & =\left( \text{\# of partitions of }n\text{ whose largest part is }\leq k\right) . \end{align*}

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theorem Nat.Partition.partsCount_sum_eq_largestPartLeq (n k : ℕ) :

∑ i ∈ Finset.range (k + 1), partsCount i n = {p : n.Partition | p.largestPart ≤ k}.card

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Theorem 2.49

Let $m\in \mathbb {N}$. Then,

\[ \sum _{n\in \mathbb {N}}\left( p_{0}\left( n\right) +p_{1}\left( n\right) +\cdots +p_{m}\left( n\right) \right) x^{n}=\prod _{k=1}^{m}\frac{1}{1-x^{k}}. \]

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theorem Nat.Partition.partsCountSum_genFun {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [T2Space R] [IsTopologicalSemiring R] (m : ℕ) :

(PowerSeries.mk fun (n : ℕ) => ∑ k ∈ Finset.range (m + 1), ↑(partsCount k n)) = ∏ k ∈ Finset.range m, ∑' (j : ℕ), PowerSeries.X ^ ((k + 1) * j)

Partition number vs. sums of divisors

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Theorem 2.61

For any positive integer $n$, let $\sigma \left( n\right) $ denote the sum of all positive divisors of $n$. (For example, $\sigma \left( 6\right) =1+2+3+6=12$ and $\sigma \left( 7\right) =1+7=8$.)

For any $n\in \mathbb {N}$, we have

\[ np\left( n\right) =\sum _{k=1}^{n}\sigma \left( k\right) p\left( n-k\right) . \]

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theorem Nat.Partition.partitionCount_divisorSum (n : ℕ) :

n * partitionCount n = ∑ k ∈ Finset.range n, divisorSum (k + 1) * partitionCount (n - k - 1)

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Theorem 2.60

For any positive integer $n$, let $\sigma _{I}\left( n\right) $ denote the sum of all positive divisors of $n$ that belong to $I$.

For any $n\in \mathbb {N}$, we have

\[ np_{I}\left( n\right) =\sum _{k=1}^{n}\sigma _{I}\left( k\right) p_{I}\left( n-k\right) . \]

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theorem Nat.Partition.partitionCount_divisorSum_restricted (I : Set ℕ) [DecidablePred fun (x : ℕ) => x ∈ I] (hI : ∀ i ∈ I, i > 0) (n : ℕ) :

n * (restricted n fun (x : ℕ) => x ∈ I).card = ∑ k ∈ Finset.range n, (∑ d ∈ (k + 1).divisors with d ∈ I, d) * (restricted (n - k - 1) fun (x : ℕ) => x ∈ I).card

Euler's pentagonal number theorem

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Definition 2.67

For any $k\in \mathbb {Z}$, define a nonnegative integer $w_{k}\in \mathbb {N}$ by

\[ w_{k}=\frac{\left( 3k-1\right) k}{2}. \]

This is called the $k$-th pentagonal number.

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def AlgebraicCombinatorics.pentagonalNumber (k : ℤ) :

eulerProduct = pentagonalSeries

AlgebraicCombinatorics.pentagonalNumber k = ((3 * k - 1) * k / 2).toNat

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Theorem 2.68 Euler’s pentagonal number theorem

We have

\[ \prod _{k=1}^{\infty }\left( 1-x^{k}\right) =\sum _{k\in \mathbb {Z}}\left( -1\right) ^{k}x^{w_{k}}. \]

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theorem AlgebraicCombinatorics.euler_pentagonal_number_theorem :

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Corollary 2.79

For each positive integer $n$, we have

\begin{align*} p\left( n\right) & =\sum _{\substack {k\in \mathbb {Z};\\ k\neq 0 }}\left( -1\right) ^{k-1}p\left( n-w_{k}\right) \\ & =p\left( n-1\right) +p\left( n-2\right) -p\left( n-5\right) -p\left( n-7\right) \\ & \ \ \ \ \ \ \ \ \ \ +p\left( n-12\right) +p\left( n-15\right) -p\left( n-22\right) -p\left( n-26\right) \pm \cdots . \end{align*}

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theorem AlgebraicCombinatorics.partition_recursive (n : ℕ) (hn : n > 0) :

↑(partitionCount n) = ∑' (k : { k : ℤ // k ≠ 0 ∧ pentagonalNumber k ≤ n }), (if (↑k).natAbs % 2 = 1 then 1 else -1) * ↑(partitionCount (n - pentagonalNumber ↑k))

The identity

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Theorem 2.80 Jacobi’s triple product identity, take 1

In the ring $\left( \mathbb {Z}\left[ z^{\pm }\right] \right) \left[ \left[ q\right] \right] $, we have

\[ \prod _{n{\gt}0}\left( \left( 1+q^{2n-1}z\right) \left( 1+q^{2n-1} z^{-1}\right) \left( 1-q^{2n}\right) \right) =\sum _{\ell \in \mathbb {Z} }q^{\ell ^{2}}z^{\ell }. \]

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theorem AlgebraicCombinatorics.jacobi_triple_product_fps' :

jacobiLHS' = jacobiRHS'

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Theorem 2.81 Jacobi’s triple product identity, take 2

Let $a$ and $b$ be two integers such that $a{\gt}0$ and $a\geq \left\vert b\right\vert $. Let $u,v\in \mathbb {Q}$ be rational numbers with $v\neq 0$. In the ring $\mathbb {Q}\left[\left[ x\right]\right] $, set $q=ux^{a}$ and $z=vx^{b}$. Then,

\[ \prod _{n{\gt}0}\left( \left( 1+q^{2n-1}z\right) \left( 1+q^{2n-1} z^{-1}\right) \left( 1-q^{2n}\right) \right) =\sum _{\ell \in \mathbb {Z} }q^{\ell ^{2}}z^{\ell }. \]

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theorem AlgebraicCombinatorics.jacobi_triple_product (a b : ℤ) (ha : a > 0) (hab : a ≥ |b|) (u v : ℚ) (hv : v ≠ 0) :

jacobiLHSEval a b u v = jacobiRHSEval a b u v

Jacobi implies Euler

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Lemma 2.119

Let $K$ be a commutative ring. Let $f$ and $g$ be two FPSs in $K[[x]]$. Assume that $f[x^{2}]=g[x^{2}]$. Then, $f=g$.

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theorem AlgebraicCombinatorics.fps_eq_of_sq_eq {R : Type u_1} [CommRing R] (f g : PowerSeries R) (h : PowerSeries.subst (PowerSeries.X ^ 2) f = PowerSeries.subst (PowerSeries.X ^ 2) g) :

f = g

Application: A recursion for the sum of divisors

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Theorem 2.130

For any positive integer $n$, let $\sigma (n)$ denote the sum of all positive divisors of $n$.

Then, for each positive integer $n$, we have

\[ \sum _{\substack {k\in \mathbb {Z};\\ w_{k}{\lt}n}}\left( -1\right) ^{k}\sigma \left( n-w_{k}\right) = \begin{cases} \left( -1\right) ^{k-1}n, & \text{if }n=w_{k}\text{ for some }k\in \mathbb {Z};\\ 0, & \text{if not.} \end{cases} \]

(Here, $w_{k}$ is the $k$-th pentagonal number, defined in Definition 2.67.)

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theorem AlgebraicCombinatorics.euler_sum_divisors_recursive (n : ℕ) (_hn : n > 0) :

∑' (k : { k : ℤ // pentagonalNumber k < n }), (-1) ^ (↑k).natAbs * ↑((ArithmeticFunction.sigma 1) (n - pentagonalNumber ↑k)) = match pentagonalNumberInverse n with | some k => (-1) ^ (k.natAbs + 1) * ↑n | none => 0

$q$-binomial coefficients

Motivation

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Proposition 2.135

For any positive integers $k$ and $\ell $, we have

\[ \left(\text{\# of partitions with }k\text{ parts and largest part }\ell \right) = \binom {k+\ell -2}{k-1}. \]

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theorem Nat.Partition.partsAndLargestCountTotal_eq (k ℓ : ℕ) (hk : k ≥ 1) (hℓ : ℓ ≥ 1) :

partsAndLargestCountTotal k ℓ = (k + ℓ - 2).choose (k - 1)

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theorem AlgebraicCombinatorics.card_partitions_with_parts_and_largest (k ℓ : ℕ) (hk : 0 < k) (hℓ : 0 < ℓ) :

(monotoneFunctions (k - 1) (ℓ - 1)).card = (k + ℓ - 2).choose (k - 1)

Definition

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Definition 2.136

Let $n\in \mathbb {N}$ and $k\in \mathbb {Z}$.

(a) The $q$-binomial coefficient (or Gaussian binomial coefficient) $\binom {n}{k}_{q}$ is defined to be the polynomial

\[ \sum _{\substack {\lambda \text{ is a partition}\\ \text{with largest part }\leq n-k\\ \text{and length }\leq k}}q^{|\lambda |}\in \mathbb {Z}[q]. \]

(b) If $a$ is any element of a ring $A$, then we set

\[ \binom {n}{k}_{a} := \binom {n}{k}_{q}[a] = \sum _{\substack {\lambda \text{ is a partition}\\ \text{with largest part }\leq n-k\\ \text{and length }\leq k}}a^{|\lambda |}\in A. \]

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noncomputable def AlgebraicCombinatorics.qBinomial {R : Type u_1} [CommSemiring R] (n k : ℕ) (q : R) :

AlgebraicCombinatorics.qBinomial n k q = if k ≤ n then ∑ f ∈ AlgebraicCombinatorics.monotoneFunctions k (n - k), q ^ ∑ i : Fin k, ↑(f i) else 0

Basic properties

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Proposition 2.140

Let $n\in \mathbb {N}$ and $k\in \mathbb {Z}$.

(a) We have

\[ \binom {n}{k}_{q}=\sum _{0\leq i_{1}\leq i_{2}\leq \cdots \leq i_{k}\leq n-k}q^{i_{1}+i_{2}+\cdots +i_{k}}. \]

Here, the sum ranges over all weakly increasing $k$-tuples $(i_{1},i_{2},\ldots ,i_{k}) \in \{ 0,1,\ldots ,n-k\} ^{k}$. If $k{\gt}n$, then this is an empty sum. If $k{\lt}0$, then this is also an empty sum.

(b) Set $\operatorname {sum}S=\sum _{s\in S}s$ for any finite set $S$ of integers. Then, we have

\[ \binom {n}{k}_{q}=\sum _{\substack {S\subseteq \{ 1,2,\ldots ,n\} ;\\ \left|S\right|=k}}q^{\operatorname {sum}S-(1+2+\cdots +k)}. \]

(c) We have

\[ \binom {n}{k}_{1}=\binom {n}{k}. \]

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theorem AlgebraicCombinatorics.qBinomial_eq_sum_increasing_tuples {R : Type u_1} [CommSemiring R] (n k : ℕ) (hk : k ≤ n) (q : R) :

qBinomial n k q = ∑ f ∈ monotoneFunctions k (n - k), q ^ ∑ i : Fin k, ↑(f i)

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theorem AlgebraicCombinatorics.QBinomialRec.qBinomial_eq_sum_subsets {R : Type u_1} [CommRing R] (q : R) (n k : ℕ) :

qBinomial q n k = ∑ S ∈ kSubsetsOfIcc n k, q ^ (finsetSumNat S - triangular k)

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theorem AlgebraicCombinatorics.qBinomial_one_eq_binomial {R : Type u_1} [CommSemiring R] (n k : ℕ) :

qBinomial n k 1 = ↑(n.choose k)

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Proposition 2.141

Let $n,k\in \mathbb {N}$ satisfy $k{\gt}n$. Then, $\binom {n}{k}_{q}=0$.

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theorem AlgebraicCombinatorics.qBinomial_zero_of_lt {R : Type u_1} [CommSemiring R] (n k : ℕ) (hk : n < k) (q : R) :

qBinomial n k q = 0

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Convention 2.143

Let $n\in \mathbb {N}$. For any $k\notin \mathbb {Z}$, we set $\binom {n}{k}_{q}:=0$.

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theorem AlgebraicCombinatorics.qBinomial_n_zero {R : Type u_1} [CommSemiring R] (n : ℕ) (q : R) :

qBinomial n 0 q = 1

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theorem AlgebraicCombinatorics.qBinomial_n_n {R : Type u_1} [CommSemiring R] (n : ℕ) (q : R) :

qBinomial n n q = 1

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Theorem 2.144

Let $n$ be a positive integer. Let $k\in \mathbb {N}$. Then:

(a) We have

\[ \binom {n}{k}_{q}=q^{n-k}\binom {n-1}{k-1}_{q}+\binom {n-1}{k}_{q}. \]

(b) We have

\[ \binom {n}{k}_{q}=\binom {n-1}{k-1}_{q}+q^{k}\binom {n-1}{k}_{q}. \]

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theorem AlgebraicCombinatorics.qBinomial_rec_left {R : Type u_1} [CommSemiring R] (n : ℕ) (hn : 0 < n) (k : ℕ) (hk : 0 < k) (q : R) :

qBinomial n k q = q ^ (n - k) * qBinomial (n - 1) (k - 1) q + qBinomial (n - 1) k q

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theorem AlgebraicCombinatorics.qBinomial_rec_right {R : Type u_1} [CommSemiring R] (n : ℕ) (hn : 0 < n) (k : ℕ) (hk : 0 < k) (q : R) :

qBinomial n k q = qBinomial (n - 1) (k - 1) q + q ^ k * qBinomial (n - 1) k q

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Theorem 2.147

Let $n,k\in \mathbb {N}$ satisfy $n\geq k$. Then:

(a) We have

\[ (1-q^{k})(1-q^{k-1})\cdots (1-q^{1})\cdot \binom {n}{k}_{q} =(1-q^{n})(1-q^{n-1})\cdots (1-q^{n-k+1}). \]

(b) We have

\[ \binom {n}{k}_{q}=\frac{(1-q^{n})(1-q^{n-1}) \cdots (1-q^{n-k+1})}{(1-q^{k})(1-q^{k-1})\cdots (1-q^{1})} \]

(in the ring $\mathbb {Z}[[q]]$ or in the field of rational functions over $\mathbb {Q}$).

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theorem AlgebraicCombinatorics.qBinomial_quot_formula {R : Type u_2} [CommRing R] (n k : ℕ) (hk : k ≤ n) (q : R) :

(∏ i ∈ Finset.range k, (1 - q ^ (i + 1))) * qBinomial n k q = ∏ i ∈ Finset.range k, (1 - q ^ (n - i))

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theorem AlgebraicCombinatorics.qBinomial_eq_prod_quot {R : Type u_2} [Field R] (n k : ℕ) (hk : k ≤ n) (q : R) (hq : ∀ i ∈ Finset.range k, q ^ (i + 1) ≠ 1) :

qBinomial n k q = ∏ i ∈ Finset.range k, (1 - q ^ (n - i)) / (1 - q ^ (i + 1))

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Definition 2.148

(a) For any $n\in \mathbb {N}$, define the $q$-integer $[n]_{q}$ to be

\[ [n]_{q}:=q^{0}+q^{1}+\cdots +q^{n-1}\in \mathbb {Z}[q]. \]

(b) For any $n\in \mathbb {N}$, define the $q$-factorial $[n]_{q}!$ to be

\[ [n]_{q}!\ :=[1]_{q}[2]_{q}\cdots [n]_{q}\in \mathbb {Z}[q]. \]

(c) As usual, if $a$ is an element of a ring $A$, then $[n]_{a}$ and $[n]_{a}!$ will mean the results of substituting $a$ for $q$ in $[n]_{q}$ and $[n]_{q}!$, respectively. Thus, explicitly, $[n]_{a}=a^{0}+a^{1}+\cdots +a^{n-1}$ and $[n]_{a}!=[1]_{a}[2]_{a}\cdots [n]_{a}$.

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def AlgebraicCombinatorics.qInt {R : Type u_1} [CommSemiring R] (n : ℕ) (q : R) :

AlgebraicCombinatorics.qInt n q = ∑ i ∈ Finset.range n, q ^ i

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def AlgebraicCombinatorics.qFactorial {R : Type u_1} [CommSemiring R] (n : ℕ) (q : R) :

AlgebraicCombinatorics.qFactorial n q = ∏ i ∈ Finset.range n, AlgebraicCombinatorics.qInt (i + 1) q

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Theorem 2.153

Let $n,k\in \mathbb {N}$ with $n\geq k$. Then,

\[ \binom {n}{k}_{q}=\frac{[n]_{q}[n-1]_{q} \cdots [n-k+1]_{q}}{[k]_{q}!}=\frac{[n]_{q}!}{[k]_{q}!\cdot [n-k]_{q}!} \]

(in the ring $\mathbb {Z}[[q]]$ or in the ring of rational functions over $\mathbb {Q}$).

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theorem AlgebraicCombinatorics.qBinomial_eq_qFactorial_quot {R : Type u_2} [Field R] (n k : ℕ) (hk : k ≤ n) (q : R) (hq : ∀ i ∈ Finset.range n, q ^ (i + 1) ≠ 1) :

qBinomial n k q = qFactorial n q / (qFactorial k q * qFactorial (n - k) q)

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Proposition 2.156

Let $n,k\in \mathbb {N}$. Then,

\[ \binom {n}{k}_{q}=\binom {n}{n-k}_{q}. \]

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theorem AlgebraicCombinatorics.qBinomial_symm {R : Type u_1} [CommSemiring R] (n k : ℕ) (hk : k ≤ n) (q : R) :

qBinomial n k q = qBinomial n (n - k) q

$q$-binomial formulas

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Theorem 2.167 1st $q$-binomial theorem

Let $K$ be a commutative ring. Let $a,b\in K$ and $n\in \mathbb {N}$. In the polynomial ring $K\left[ q\right] $, we have

\[ \left( aq^{0}+b\right) \left( aq^{1}+b\right) \cdots \left( aq^{n-1}+b\right) =\sum _{k=0}^{n}q^{k\left( k-1\right) /2}\binom {n}{k}_{q}a^{k}b^{n-k}. \]

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theorem AlgebraicCombinatorics.QBinomialRec.qBinomial_first_theorem {K : Type u_1} [CommRing K] (a b q : K) (n : ℕ) :

∏ i ∈ Finset.range n, (a * q ^ i + b) = ∑ k ∈ Finset.range (n + 1), q ^ (k * (k - 1) / 2) * qBinomial q n k * a ^ k * b ^ (n - k)

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Lemma 2.164

Let $L$ be a commutative ring. Let $n\in \mathbb {N}$. Let $\left[ n\right] $ denote the set $\left\{ 1,2,\ldots ,n\right\} $. Let $a_{1},a_{2},\ldots ,a_{n}$ be $n$ elements of $L$. Let $b_{1},b_{2},\ldots ,b_{n}$ be $n$ further elements of $L$. Then,

\begin{equation} \prod _{i=1}^{n}\left( a_{i}+b_{i}\right) =\sum _{S\subseteq \left[ n\right] }\left( \prod _{i\in S}a_{i}\right) \left( \prod _{i\in \left[ n\right] \setminus S}b_{i}\right) . \label{eq.lem.prodrule.sum-ai-plus-bi.eq} \end{equation}

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theorem AlgebraicCombinatorics.QBinomialRec.prod_add_eq_sum_over_subsets {L : Type u_1} [CommRing L] {n : ℕ} (a b : Fin n → L) :

∏ i : Fin n, (a i + b i) = ∑ S : Finset (Fin n), (∏ i ∈ S, a i) * ∏ i ∈ Sᶜ, b i

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Theorem 2.170 2nd $q$-binomial theorem, aka Potter’s binomial theorem

Let $L$ be a commutative ring. Let $\omega \in L$. Let $A$ be a noncommutative $L$-algebra. Let $a,b\in A$ be such that $ba=\omega ab$. Then,

\[ \left( a+b\right) ^{n}=\sum _{k=0}^{n}\binom {n}{k}_{\omega }a^{k}b^{n-k}. \]

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theorem AlgebraicCombinatorics.QBinomialRec.qBinomial_second_theorem {L : Type u_1} [CommRing L] {A : Type u_2} [Ring A] [Algebra L A] (ω : L) (a b : A) (hab : b * a = ω • (a * b)) (n : ℕ) :

(a + b) ^ n = ∑ k ∈ Finset.range (n + 1), qBinomial ω n k • (a ^ k * b ^ (n - k))

Counting subspaces of vector spaces

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Theorem 2.176

Let $F$ be a finite field. Let $n,k\in \mathbb {N}$. Let $V$ be an $n$-dimensional $F$-vector space. Then,

\[ \binom {n}{k}_{\left\vert F\right\vert }=\left( \text{\# of }k\text{-dimensional vector subspaces of }V\right) . \]

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theorem AlgebraicCombinatorics.QBinomialRec.qBinomial_subspace_count {F : Type u_1} [Field F] [Fintype F] {V : Type u_2} [AddCommGroup V] [Module F V] [Module.Finite F V] (n k : ℕ) (hn : Module.finrank F V = n) :

qBinomial (↑(Fintype.card F)) n k = ↑(Nat.card { W : Submodule F V // Module.finrank F ↥W = k })

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Lemma 2.173

Let $F$ be a field. Let $V$ be an $F$-vector space. Let $\left( v_{1},v_{2},\ldots ,v_{k}\right) $ be a $k$-tuple of vectors in $V$. Then, $\left( v_{1},v_{2},\ldots ,v_{k}\right) $ is linearly independent if and only if each $i\in \left\{ 1,2,\ldots ,k\right\} $ satisfies $v_{i}\notin \operatorname {span}\left( v_{1},v_{2},\ldots ,v_{i-1}\right) $ (where the span $\operatorname {span}\left( {}\right) $ of an empty list is understood to be the set $\left\{ \mathbf{0}\right\} $ consisting only of the zero vector $\mathbf{0}$). In other words, $\left( v_{1},v_{2},\ldots ,v_{k}\right) $ is linearly independent if and only if we have

\begin{align*} v_{1} & \notin \underbrace{\operatorname {span}\left( {}\right) }_{=\left\{ \mathbf{0}\right\} }\ \ \ \ \ \ \ \ \ \ \text{and}\\ v_{2} & \notin \operatorname {span}\left( v_{1}\right) \ \ \ \ \ \ \ \ \ \ \text{and}\\ v_{3} & \notin \operatorname {span}\left( v_{1},v_{2}\right) \ \ \ \ \ \ \ \ \ \ \text{and}\\ & \cdots \ \ \ \ \ \ \ \ \ \ \text{and}\\ v_{k} & \notin \operatorname {span}\left( v_{1},v_{2},\ldots ,v_{k-1}\right) . \end{align*}

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theorem AlgebraicCombinatorics.QBinomialRec.linearIndependent_iff_not_mem_span_of_lt {F : Type u_1} [Field F] {V : Type u_2} [AddCommGroup V] [Module F V] {k : ℕ} (v : Fin k → V) :

LinearIndependent F v ↔ ∀ (i : Fin k), v i ∉ Submodule.span F (v '' {j : Fin k | ↑j < ↑i})

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Lemma 2.174

Let $F$ be a finite field. Let $n,k\in \mathbb {N}$. Let $V$ be an $n$-dimensional $F$-vector space. Then,

\begin{align*} & \left( \text{\# of linearly independent }k\text{-tuples of vectors in }V\right) \\ & =\left( \left\vert F\right\vert ^{n}-\left\vert F\right\vert ^{0}\right) \left( \left\vert F\right\vert ^{n}-\left\vert F\right\vert ^{1}\right) \cdots \left( \left\vert F\right\vert ^{n}-\left\vert F\right\vert ^{k-1}\right) =\prod _{i=0}^{k-1}\left( \left\vert F\right\vert ^{n}-\left\vert F\right\vert ^{i}\right) . \end{align*}

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theorem AlgebraicCombinatorics.QBinomialRec.card_linearIndependent_tuples {F : Type u_1} [Field F] [Fintype F] {V : Type u_2} [AddCommGroup V] [Module F V] [Module.Finite F V] (n k : ℕ) (hn : Module.finrank F V = n) :

Nat.card { v : Fin k → V // LinearIndependent F v } = ∏ i ∈ Finset.range k, (Fintype.card F ^ n - Fintype.card F ^ i)

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Lemma 2.175 Multijection principle

Let $A$ and $B$ be two finite sets. Let $m\in \mathbb {N}$. Let $f:A\rightarrow B$ be any map. Assume that each $b\in B$ has exactly $m$ preimages under $f$ (that is, for each $b\in B$, there are exactly $m$ many elements $a\in A$ such that $f\left( a\right) =b$). Then,

\[ \left\vert A\right\vert =m\cdot \left\vert B\right\vert . \]

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theorem AlgebraicCombinatorics.QBinomialRec.card_eq_mul_of_fibers {A : Type u_3} {B : Type u_4} [Fintype A] [Fintype B] [DecidableEq B] (f : A → B) (m : ℕ) (hf : ∀ (b : B), Fintype.card { a : A // f a = b } = m) :

Fintype.card A = m * Fintype.card B

Limits of $q$-binomial coefficients

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Proposition 2.182

Let $k\in \mathbb {N}$ be fixed. Then,

\[ \lim _{n\rightarrow \infty }\binom {n}{k}_{q}=\sum _{n\in \mathbb {N}}\left( p_{0}\left( n\right) +p_{1}\left( n\right) +\cdots +p_{k}\left( n\right) \right) q^{n}=\prod _{i=1}^{k}\frac{1}{1-q^{i}}. \]

(See Definition 2.10 (a) for the meaning of $p_{i}\left( n\right) $.)

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theorem AlgebraicCombinatorics.QBinomialRec.qBinomial_limit (k : ℕ) :

Filter.Tendsto (fun (n : ℕ) => qBinomial PowerSeries.X n k) Filter.atTop (nhds (∏ i ∈ Finset.range k, (1 - PowerSeries.X ^ (i + 1))⁻¹))

Basic definitions

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Definition 3.2

Let $X$ be a set.

(a) A permutation of $X$ means a bijection from $X$ to $X$.

(b) It is known that the set of all permutations of $X$ is a group under composition. This group is called the symmetric group of $X$, and is denoted by $S_X$. Its neutral element is the identity map $\operatorname {id}_X : X \to X$. Its size is $|X|!$ when $X$ is finite.

(c) As usual in group theory, we will write $\alpha \beta $ for the composition $\alpha \circ \beta $ when $\alpha , \beta \in S_X$. This is the map that sends each $x \in X$ to $\alpha (\beta (x))$.

(d) If $\alpha \in S_X$ and $i \in \mathbb {Z}$, then $\alpha ^i$ shall denote the $i$-th power of $\alpha $ in the group $S_X$. Note that $\alpha ^i = \underbrace{\alpha \circ \alpha \circ \cdots \circ \alpha }_{i\text{ times}}$ if $i \ge 0$. Also, $\alpha ^0 = \operatorname {id}_X$. Also, $\alpha ^{-1}$ is the inverse of $\alpha $ in the group $S_X$, that is, the inverse of the map $\alpha $.

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abbrev AlgebraicCombinatorics.Permutation (X : Type u_1) :

Type u_1

AlgebraicCombinatorics.Permutation X = Equiv.Perm X

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abbrev AlgebraicCombinatorics.SymmetricGroup (X : Type u_1) :

Type u_1

AlgebraicCombinatorics.SymmetricGroup X = Equiv.Perm X

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theorem AlgebraicCombinatorics.symmetricGroup_card (X : Type u_1) [Fintype X] [DecidableEq X] :

Fintype.card (Equiv.Perm X) = (Fintype.card X).factorial

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theorem AlgebraicCombinatorics.perm_mul_apply {X : Type u_1} (α β : Equiv.Perm X) (x : X) :

(α * β) x = α (β x)

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theorem AlgebraicCombinatorics.perm_pow_apply {X : Type u_1} (α : Equiv.Perm X) (n : ℕ) (x : X) :

(α ^ n) x = (⇑α)^[n] x

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theorem AlgebraicCombinatorics.perm_zpow_neg_one {X : Type u_1} (α : Equiv.Perm X) :

α ^ (-1) = α⁻¹

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Definition 3.3

Let $n \in \mathbb {Z}$. Then, $[n]$ shall mean the set $\{ 1, 2, \ldots , n\} $. This is an $n$-element set if $n \ge 0$, and is an empty set if $n \le 0$.

The symmetric group $S_{[n]}$ (consisting of all permutations of $[n]$) will be denoted $S_n$ and called the $n$-th symmetric group. Its size is $n!$ (when $n \ge 0$).

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abbrev AlgebraicCombinatorics.bracketN (n : ℕ) :

AlgebraicCombinatorics.bracketN n = Fin n

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abbrev AlgebraicCombinatorics.Sn (n : ℕ) :

AlgebraicCombinatorics.Sn n = Equiv.Perm (Fin n)

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theorem AlgebraicCombinatorics.sn_card (n : ℕ) :

Fintype.card (Sn n) = n.factorial

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Proposition 3.4

Let $X$ and $Y$ be two sets, and let $f : X \to Y$ be a bijection. Then, for each permutation $\sigma $ of $X$, the map $f \circ \sigma \circ f^{-1} : Y \to Y$ is a permutation of $Y$. Furthermore, the map

\begin{align*} S_f : S_X & \to S_Y, \\ \sigma & \mapsto f \circ \sigma \circ f^{-1} \end{align*}

is a group isomorphism; thus, we obtain $S_X \cong S_Y$.

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def AlgebraicCombinatorics.symmetricGroup_conj_iso {X : Type u_1} {Y : Type u_2} (f : X ≃ Y) :

Equiv.Perm X ≃* Equiv.Perm Y

AlgebraicCombinatorics.symmetricGroup_conj_iso f = f.permCongrHom

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theorem AlgebraicCombinatorics.symmetricGroup_iso_of_equiv {X : Type u_1} {Y : Type u_2} (f : X ≃ Y) :

Nonempty (Equiv.Perm X ≃* Equiv.Perm Y)

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Definition 3.5

Let $n \in \mathbb {N}$ and $\sigma \in S_n$. We introduce three notations for $\sigma $:

(a) A two-line notation of $\sigma $ means a $2 \times n$-array

\[ \begin{pmatrix} p_1 & p_2 & \cdots & p_n \\ \sigma (p_1) & \sigma (p_2) & \cdots & \sigma (p_n) \end{pmatrix}, \]

where the entries $p_1, p_2, \ldots , p_n$ of the top row are the $n$ elements of $[n]$ in some order. Commonly, we pick $p_i = i$.

(b) The one-line notation (short, OLN) of $\sigma $ means the $n$-tuple $(\sigma (1), \sigma (2), \ldots , \sigma (n))$.

(c) The cycle digraph of $\sigma $ is the directed graph with vertices $1, 2, \ldots , n$ and arcs $i \to \sigma (i)$ for all $i \in [n]$.

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def AlgebraicCombinatorics.twoLineNotation {n : ℕ} (σ : Sn n) :

List (Fin n × Fin n)

AlgebraicCombinatorics.twoLineNotation σ = List.map (fun (i : Fin n) => (i, σ i)) (List.finRange n)

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def AlgebraicCombinatorics.oneLineNotation {n : ℕ} (σ : Sn n) :

List (Fin n)

AlgebraicCombinatorics.oneLineNotation σ = List.map (⇑σ) (List.finRange n)

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def AlgebraicCombinatorics.cycleDigraph {n : ℕ} (σ : Sn n) :

SimpleGraph (Fin n)

AlgebraicCombinatorics.cycleDigraph σ = { Adj := fun (i j : Fin n) => i ≠ j ∧ (σ i = j ∨ σ j = i), symm := ⋯, loopless := ⋯ }

Transpositions

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Definition 3.7

Let $i$ and $j$ be two distinct elements of a set $X$.

Then, the transposition $t_{i,j}$ is the permutation of $X$ that sends $i$ to $j$, sends $j$ to $i$, and leaves all other elements of $X$ unchanged.

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abbrev AlgebraicCombinatorics.transposition {X : Type u_1} [DecidableEq X] (i j : X) :

Equiv.Perm X

AlgebraicCombinatorics.transposition i j = Equiv.swap i j

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theorem AlgebraicCombinatorics.transposition_apply_left {X : Type u_1} [DecidableEq X] (i j : X) :

(transposition i j) i = j

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theorem AlgebraicCombinatorics.transposition_apply_right {X : Type u_1} [DecidableEq X] (i j : X) :

(transposition i j) j = i

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theorem AlgebraicCombinatorics.transposition_apply_of_ne_of_ne {X : Type u_1} [DecidableEq X] {i j x : X} (hi : x ≠ i) (hj : x ≠ j) :

(transposition i j) x = x

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Definition 3.12

Let $n \in \mathbb {N}$ and $i \in [n-1]$. Then, the simple transposition $s_i$ is defined by

\[ s_i := t_{i, i+1} \in S_n. \]

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def AlgebraicCombinatorics.simpleTransposition {n : ℕ} (i : Fin (n - 1)) :

Sn n

AlgebraicCombinatorics.simpleTransposition i = Equiv.swap ⟨↑i, ⋯⟩ ⟨↑i + 1, ⋯⟩

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theorem AlgebraicCombinatorics.simpleTransposition_eq_transposition {n : ℕ} (i : Fin (n - 1)) :

simpleTransposition i = transposition ⟨↑i, ⋯⟩ ⟨↑i + 1, ⋯⟩

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theorem AlgebraicCombinatorics.simpleTransposition_apply_self {n : ℕ} (i : Fin (n - 1)) :

(simpleTransposition i) ⟨↑i, ⋯⟩ = ⟨↑i + 1, ⋯⟩

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theorem AlgebraicCombinatorics.simpleTransposition_apply_succ {n : ℕ} (i : Fin (n - 1)) :

(simpleTransposition i) ⟨↑i + 1, ⋯⟩ = ⟨↑i, ⋯⟩

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theorem AlgebraicCombinatorics.simpleTransposition_apply_of_ne {n : ℕ} (i : Fin (n - 1)) (k : Fin n) (hi : ↑k ≠ ↑i) (hi1 : ↑k ≠ ↑i + 1) :

(simpleTransposition i) k = k

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Proposition 3.17

Let $n \in \mathbb {N}$.

(a) We have $s_i^2 = \operatorname {id}$ for all $i \in [n-1]$. In other words, we have $s_i = s_i^{-1}$ for all $i \in [n-1]$.

(b) We have $s_i s_j = s_j s_i$ for any $i, j \in [n-1]$ with $|i - j| {\gt} 1$.

(c) We have $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$ for any $i \in [n-2]$.

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theorem AlgebraicCombinatorics.simpleTransposition_sq_eq_one {n : ℕ} (i : Fin (n - 1)) :

simpleTransposition i * simpleTransposition i = 1

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theorem AlgebraicCombinatorics.simpleTransposition_comm {n : ℕ} (i j : Fin (n - 1)) (h : ↑i + 1 < ↑j ∨ ↑j + 1 < ↑i) :

simpleTransposition i * simpleTransposition j = simpleTransposition j * simpleTransposition i

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theorem AlgebraicCombinatorics.simpleTransposition_braid {n : ℕ} (i : Fin (n - 2)) :

have i' := ⟨↑i, ⋯⟩; have i1 := ⟨↑i + 1, ⋯⟩; simpleTransposition i' * simpleTransposition i1 * simpleTransposition i' = simpleTransposition i1 * simpleTransposition i' * simpleTransposition i1

Cycles

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Definition 3.18

Let $X$ be a set. Let $i_1, i_2, \ldots , i_k$ be $k$ distinct elements of $X$. Then,

\[ \operatorname {cyc}_{i_1, i_2, \ldots , i_k} \]

means the permutation of $X$ that sends

\begin{align*} & i_1 \text{ to } i_2, \\ & i_2 \text{ to } i_3, \\ & \ldots , \\ & i_{k-1} \text{ to } i_k, \\ & i_k \text{ to } i_1 \end{align*}

and leaves all other elements of $X$ unchanged. In other words, $\operatorname {cyc}_{i_1, i_2, \ldots , i_k}$ means the permutation of $X$ that satisfies

\[ \operatorname {cyc}_{i_1, \ldots , i_k}(p) = \begin{cases} i_{j+1}, & \text{if } p = i_j \text{ for some } j \in \{ 1, 2, \ldots , k\} ;\\ p, & \text{otherwise} \end{cases} \]

for every $p \in X$, where $i_{k+1}$ means $i_1$.

This permutation is called a $k$-cycle.

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def AlgebraicCombinatorics.cyc {X : Type u_1} [DecidableEq X] (l : List X) :

Equiv.Perm X

AlgebraicCombinatorics.cyc l = l.formPerm

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theorem AlgebraicCombinatorics.cyc_apply_getElem {X : Type u_1} [DecidableEq X] {l : List X} (hl : l.Nodup) (j : ℕ) (hj : j < l.length) :

(cyc l) l[j] = l[(j + 1) % l.length]

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theorem AlgebraicCombinatorics.cyc_apply_of_not_mem {X : Type u_1} [DecidableEq X] {l : List X} {x : X} (h : x ∉ l) :

(cyc l) x = x

Involutions

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Definition 3.27

Let $X$ be a set. An involution of $X$ means a map $f : X \to X$ that satisfies $f \circ f = \operatorname {id}$. Clearly, an involution is always a permutation, and equals its own inverse.

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def AlgebraicCombinatorics.IsInvolution {X : Type u_1} (σ : Equiv.Perm X) :

AlgebraicCombinatorics.IsInvolution σ = (σ * σ = 1)

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theorem AlgebraicCombinatorics.isInvolution_iff_eq_inv {X : Type u_1} {σ : Equiv.Perm X} :

IsInvolution σ ↔ σ⁻¹ = σ

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theorem AlgebraicCombinatorics.isInvolution_iff_involutive {X : Type u_1} {σ : Equiv.Perm X} :

IsInvolution σ ↔ Function.Involutive ⇑σ

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theorem AlgebraicCombinatorics.isInvolution_iff_forall {X : Type u_1} {σ : Equiv.Perm X} :

IsInvolution σ ↔ ∀ (x : X), σ (σ x) = x

Inversions, length and Lehmer codes

Inversions and lengths

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Definition 3.35

Let $n\in \mathbb {N}$ and $\sigma \in S_{n}$.

(a) An inversion of $\sigma $ means a pair $\left(i,j\right)$ of elements of $\left[n\right]$ such that $i{\lt}j$ and $\sigma \left(i\right) {\gt}\sigma \left(j\right)$.

(b) The length (also known as the Coxeter length) of $\sigma $ is the # of inversions of $\sigma $. It is called $\ell \left( \sigma \right)$.

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def AlgebraicCombinatorics.Perm.inv {n : ℕ} (σ : Equiv.Perm (Fin n)) :

Finset (Fin n × Fin n)

AlgebraicCombinatorics.Perm.inv σ = {p : Fin n × Fin n | p.1 < p.2 ∧ σ p.1 > σ p.2}

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def AlgebraicCombinatorics.Perm.invCount {n : ℕ} (σ : Equiv.Perm (Fin n)) :

AlgebraicCombinatorics.Perm.invCount σ = (AlgebraicCombinatorics.Perm.inv σ).card

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Proposition 3.37

Let $n\in \mathbb {N}$.

(a) For any $\sigma \in S_{n}$, we have $\ell \left(\sigma \right) \in \left\{ 0,1,\ldots ,\binom {n}{2}\right\} $.

(b) We have

\[ \left(\text{\# of }\sigma \in S_{n}\text{ with }\ell \left(\sigma \right) =0\right) = 1. \]

Indeed, the only permutation $\sigma \in S_{n}$ with $\ell \left( \sigma \right)=0$ is the identity map $\operatorname {id}$.

(c) We have

\[ \left(\text{\# of }\sigma \in S_{n}\text{ with }\ell \left(\sigma \right) =\binom {n}{2}\right) = 1. \]

Indeed, the only permutation $\sigma \in S_{n}$ with $\ell \left( \sigma \right)=\binom {n}{2}$ is the permutation with OLN $n\left( n-1\right)\left(n-2\right)\cdots 21$, often called $w_{0}$.

(d) If $n\geq 1$, then

\[ \left(\text{\# of }\sigma \in S_{n}\text{ with }\ell \left(\sigma \right) =1\right) = n-1. \]

Indeed, the only permutations $\sigma \in S_{n}$ with $\ell \left( \sigma \right)=1$ are the simple transpositions $s_{i}$ with $i\in \left[ n-1\right]$.

(e) If $n\geq 2$, then

\[ \left(\text{\# of }\sigma \in S_{n}\text{ with }\ell \left(\sigma \right) =2\right) = \frac{\left(n-2\right)\left(n+1\right)}{2}. \]

Indeed, the only permutations $\sigma \in S_{n}$ with $\ell \left( \sigma \right)=2$ are the products $s_{i}s_{j}$ with $1\leq i{\lt}j{\lt}n$ as well as the products $s_{i}s_{i-1}$ with $i\in \left\{ 2,3,\ldots ,n-1\right\} $. If $n\geq 2$, then there are $\frac{\left(n-2\right)\left(n+1\right)}{2}$ such products (and they are all distinct).

(f) For any $k\in \mathbb {Z}$, we have

\[ \left(\text{\# of }\sigma \in S_{n}\text{ with }\ell \left(\sigma \right) =k\right) = \left(\text{\# of }\sigma \in S_{n}\text{ with }\ell \left( \sigma \right)=\binom {n}{2}-k\right). \]

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theorem AlgebraicCombinatorics.Perm.card_invCount_symm {n : ℕ} (k : ℤ) :

{σ : Equiv.Perm (Fin n) | ↑(invCount σ) = k}.card = {σ : Equiv.Perm (Fin n) | ↑(invCount σ) = ↑(n.choose 2) - k}.card

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Proposition 3.53

Let $n\in \mathbb {N}$. Then,

\begin{align*} \sum _{\sigma \in S_{n}} x^{\ell \left(\sigma \right)} & = \prod _{i=1}^{n-1}\left(1+x+x^{2}+\cdots +x^{i}\right) \\ & = \left(1+x\right)\left(1+x+x^{2}\right)\left(1+x+x^{2}+x^{3}\right) \cdots \left(1+x+x^{2}+\cdots +x^{n-1}\right) \\ & = \left[n\right]_{x}!. \end{align*}

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theorem AlgebraicCombinatorics.length_generating_function (n x : ℕ) :

∑ σ : Equiv.Perm (Fin n), x ^ Perm.invCount σ = ∏ i ∈ Finset.range n, ∑ j ∈ Finset.range (i + 1), x ^ j

Lehmer codes

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Definition 3.45

Let $n\in \mathbb {N}$.

(a) For each $\sigma \in S_{n}$ and $i\in \left[n\right]$, we set

\begin{align*} \ell _{i}\left(\sigma \right) := & \left(\text{\# of all }j\in \left[n\right] \text{ satisfying }i{\lt}j\text{ and }\sigma \left(i\right) {\gt}\sigma \left(j\right)\right) \\ = & \left(\text{\# of all }j\in \left\{ i+1,i+2,\ldots ,n\right\} \text{ such that }\sigma \left(i\right){\gt}\sigma \left(j\right)\right). \end{align*}

(b) For each $m\in \mathbb {Z}$, we let $\left[m\right]_{0}$ denote the set $\left\{ 0,1,\ldots ,m\right\} $. (This is an empty set when $m{\lt}0$.)

(c) We let $H_{n}$ denote the set

\begin{align*} & \left[n-1\right]_{0}\times \left[n-2\right]_{0}\times \cdots \times \left[n-n\right]_{0}\\ & = \left\{ \left(j_{1},j_{2},\ldots ,j_{n}\right)\in \mathbb {N}^{n} \ \mid \ j_{i}\leq n-i\text{ for each }i\in \left[n\right]\right\} . \end{align*}

This set $H_{n}$ has size $n!$.

(d) Each $\sigma \in S_n$ and each $i\in [n]$ satisfy $\ell _i(\sigma ) \in \{ 0,1,\ldots ,n-i\} = [n-i]_0$.

(e) We define the map

\begin{align*} L:S_{n} & \rightarrow H_{n},\\ \sigma & \mapsto \left(\ell _{1}\left(\sigma \right),\ell _{2}\left( \sigma \right),\ldots ,\ell _{n}\left(\sigma \right)\right). \end{align*}

If $\sigma \in S_{n}$ is a permutation, then the $n$-tuple $L\left(\sigma \right)$ is called the Lehmer code (or just the code) of $\sigma $.

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def AlgebraicCombinatorics.Perm.lehmerCode {n : ℕ} (σ : Equiv.Perm (Fin n)) :

Fin n → ℕ

AlgebraicCombinatorics.Perm.lehmerCode σ i = AlgebraicCombinatorics.Perm.lehmerEntry σ i

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theorem AlgebraicCombinatorics.Perm.lehmerCode_mem_lehmerCodeSet {n : ℕ} (σ : Equiv.Perm (Fin n)) :

lehmerCode σ ∈ lehmerCodeSet n

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Proposition 3.46

Let $n\in \mathbb {N}$ and $\sigma \in S_{n}$. Then, $\ell \left(\sigma \right)=\ell _{1}\left(\sigma \right)+\ell _{2}\left( \sigma \right)+\cdots +\ell _{n}\left(\sigma \right)$.

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theorem AlgebraicCombinatorics.Perm.invCount_eq_sum_lehmerCode {n : ℕ} (σ : Equiv.Perm (Fin n)) :

invCount σ = ∑ i : Fin n, lehmerEntry σ i

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Theorem 3.47

Let $n\in \mathbb {N}$. Then, the map $L:S_{n}\rightarrow H_{n}$ is a bijection.

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theorem AlgebraicCombinatorics.Perm.lehmerCode_bijective (n : ℕ) :

Function.Bijective fun (σ : Equiv.Perm (Fin n)) => ⟨lehmerCode σ, ⋯⟩

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theorem AlgebraicCombinatorics.Perm.lehmerCode_injective (n : ℕ) :

Function.Injective fun (σ : Equiv.Perm (Fin n)) => lehmerCode σ

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Definition 3.50

Let $\left(a_{1},a_{2},\ldots ,a_{n}\right)$ and $\left(b_{1},b_{2},\ldots ,b_{n}\right)$ be two $n$-tuples of integers. We say that

\[ \left(a_{1},a_{2},\ldots ,a_{n}\right) {\lt}_{\operatorname {lex}}\left( b_{1},b_{2},\ldots ,b_{n}\right) \]

if and only if

there exists some $k\in \left[n\right]$ such that $a_{k}\neq b_{k}$, and
the smallest such $k$ satisfies $a_{k}{\lt}b_{k}$.

The relation ${\lt}_{\operatorname {lex}}$ is a strict total order on $\mathbb {Z}^n$ (the lexicographic order).

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theorem AlgebraicCombinatorics.lexLt_irrefl {n : ℕ} (a : Fin n → ℤ) :

¬lexLt a a

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theorem AlgebraicCombinatorics.lexLt_trans {n : ℕ} {a b c : Fin n → ℤ} (hab : lexLt a b) (hbc : lexLt b c) :

lexLt a c

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theorem AlgebraicCombinatorics.lexLt_asymm {n : ℕ} {a b : Fin n → ℤ} (hab : lexLt a b) :

¬lexLt b a

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theorem AlgebraicCombinatorics.lexLt_strictTotalOrder {n : ℕ} (a b : Fin n → ℤ) :

a = b ∧ ¬lexLt a b ∧ ¬lexLt b a ∨ a ≠ b ∧ lexLt a b ∧ ¬lexLt b a ∨ a ≠ b ∧ ¬lexLt a b ∧ lexLt b a

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Proposition 3.51

If $\mathbf{a}$ and $\mathbf{b}$ are two distinct $n$-tuples of integers, then we have either $\mathbf{a} {\lt}_{\operatorname {lex}}\mathbf{b}$ or $\mathbf{b}{\lt}_{\operatorname {lex}} \mathbf{a}$.

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theorem AlgebraicCombinatorics.lexLt_trichotomous {n : ℕ} (a b : Fin n → ℤ) (hab : a ≠ b) :

lexLt a b ∨ lexLt b a

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Proposition 3.52

Let $\sigma \in S_{n}$ and $\tau \in S_{n}$ be such that

\[ \left(\sigma \left(1\right),\sigma \left(2\right),\ldots ,\sigma \left( n\right)\right) {\lt}_{\operatorname {lex}}\left(\tau \left(1\right) ,\tau \left(2\right),\ldots ,\tau \left(n\right)\right). \]

Then,

\[ \left(\ell _{1}\left(\sigma \right),\ell _{2}\left(\sigma \right) ,\ldots ,\ell _{n}\left(\sigma \right)\right) {\lt}_{\operatorname {lex}}\left( \ell _{1}\left(\tau \right),\ell _{2}\left(\tau \right),\ldots ,\ell _{n}\left(\tau \right)\right). \]

(In other words, $L\left(\sigma \right) {\lt}_{\operatorname {lex}} L\left( \tau \right)$.)

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theorem AlgebraicCombinatorics.Perm.lehmerCode_preserves_lexLt {n : ℕ} (σ τ : Equiv.Perm (Fin n)) (h : lexLt (fun (i : Fin n) => ↑↑(σ i)) fun (i : Fin n) => ↑↑(τ i)) :

lexLt (fun (i : Fin n) => ↑(lehmerEntry σ i)) fun (i : Fin n) => ↑(lehmerEntry τ i)

More about lengths and simples

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Proposition 3.54

Let $n \in \mathbb {N}$ and $\sigma \in S_n$. Then, $\ell (\sigma ^{-1}) = \ell (\sigma )$.

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theorem Equiv.Perm.length_inv {n : ℕ} (σ : Perm (Fin n)) :

σ⁻¹.length = σ.length

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Lemma 3.56 Single swap lemma

Let $n \in \mathbb {N}$, $\sigma \in S_n$ and $k \in [n-1]$. Then:

(a) We have

\[ \ell (\sigma s_k) = \begin{cases} \ell (\sigma ) + 1, & \text{if } \sigma (k) {\lt} \sigma (k+1); \\ \ell (\sigma ) - 1, & \text{if } \sigma (k) {\gt} \sigma (k+1). \end{cases} \]

(b) We have

\[ \ell (s_k \sigma ) = \begin{cases} \ell (\sigma ) + 1, & \text{if } \sigma ^{-1}(k) {\lt} \sigma ^{-1}(k+1); \\ \ell (\sigma ) - 1, & \text{if } \sigma ^{-1}(k) {\gt} \sigma ^{-1}(k+1). \end{cases} \]

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theorem Equiv.Perm.length_mul_swap_right {n : ℕ} (σ : Perm (Fin n)) (k : Fin (n - 1)) (h : n > 1) :

(σ * simpleTransposition k).length = if σ (Fin.castLE ⋯ k.castSucc) < σ (Fin.castLE ⋯ k.succ) then σ.length + 1 else σ.length - 1

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theorem Equiv.Perm.length_mul_swap_left {n : ℕ} (σ : Perm (Fin n)) (k : Fin (n - 1)) (h : n > 1) :

(simpleTransposition k * σ).length = if σ⁻¹ (Fin.castLE ⋯ k.castSucc) < σ⁻¹ (Fin.castLE ⋯ k.succ) then σ.length + 1 else σ.length - 1

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Proposition 3.59

Let $n \in \mathbb {N}$ and $\sigma \in S_n$. Let $i$ and $j$ be two elements of $[n]$ such that $i {\lt} j$. Then:

(a) We have $\ell (\sigma t_{i,j}) {\lt} \ell (\sigma )$ if $\sigma (i) {\gt} \sigma (j)$. We have $\ell (\sigma t_{i,j}) {\gt} \ell (\sigma )$ if $\sigma (i) {\lt} \sigma (j)$.

(b) We have

\[ \ell (\sigma t_{i,j}) = \begin{cases} \ell (\sigma ) - 2|Q| - 1, & \text{if } \sigma (i) {\gt} \sigma (j); \\ \ell (\sigma ) + 2|R| + 1, & \text{if } \sigma (i) {\lt} \sigma (j), \end{cases} \]

where

\begin{align*} Q & = \{ k \in \{ i+1, i+2, \ldots , j-1\} \mid \sigma (i) {\gt} \sigma (k) {\gt} \sigma (j)\} \quad \text{and} \\ R & = \{ k \in \{ i+1, i+2, \ldots , j-1\} \mid \sigma (i) {\lt} \sigma (k) {\lt} \sigma (j)\} . \end{align*}

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theorem Equiv.Perm.length_mul_transposition_compare {n : ℕ} (σ : Perm (Fin n)) (i j : Fin n) (hij : i < j) :

(σ j < σ i → (σ * swap i j).length < σ.length) ∧ (σ i < σ j → σ.length < (σ * swap i j).length)

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theorem Equiv.Perm.length_mul_transposition {n : ℕ} (σ : Perm (Fin n)) (i j : Fin n) (hij : i < j) :

(σ * swap i j).length = if σ j < σ i then σ.length - 2 * (σ.setQ i j).card - 1 else σ.length + 2 * (σ.setR i j).card + 1

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Theorem 3.81 First reduced word theorem for the symmetric group

Let $n \in \mathbb {N}$ and $\sigma \in S_n$. Then:

(a) We can write $\sigma $ as a composition (i.e., product) of $\ell (\sigma )$ simples.

(b) The number $\ell (\sigma )$ is the smallest $p \in \mathbb {N}$ such that we can write $\sigma $ as a composition of $p$ simples.

Keep in mind: The composition of $0$ simples is $\mathrm{id}$, since $\mathrm{id}$ is the neutral element of the group $S_n$.

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theorem Equiv.Perm.exists_reduced_word {n : ℕ} (σ : Perm (Fin n)) :

∃ (w : Word n), IsReduced w ∧ wordProd w = σ

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theorem Equiv.Perm.length_le_word_length {n : ℕ} (w : Word n) :

(wordProd w).length ≤ List.length w

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Corollary 3.85

Let $n \in \mathbb {N}$.

(a) We have $\ell (\sigma \tau ) \equiv \ell (\sigma ) + \ell (\tau ) \pmod{2}$ for all $\sigma , \tau \in S_n$.

(b) We have $\ell (\sigma \tau ) \leq \ell (\sigma ) + \ell (\tau )$ for all $\sigma , \tau \in S_n$.

(c) Let $k_1, k_2, \ldots , k_q \in [n-1]$, and let $\sigma = s_{k_1} s_{k_2} \cdots s_{k_q}$. Then $q \geq \ell (\sigma )$ and $q \equiv \ell (\sigma ) \pmod{2}$.

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theorem Equiv.Perm.length_mul_mod_two {n : ℕ} (σ τ : Perm (Fin n)) :

(σ * τ).length % 2 = (σ.length + τ.length) % 2

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theorem Equiv.Perm.length_mul_le {n : ℕ} (σ τ : Perm (Fin n)) :

(σ * τ).length ≤ σ.length + τ.length

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theorem Equiv.Perm.word_length_ge_and_parity {n : ℕ} (w : Word n) :

List.length w ≥ (wordProd w).length ∧ List.length w % 2 = (wordProd w).length % 2

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Corollary 3.89

Let $n \in \mathbb {N}$. Then, the group $S_n$ is generated by the simples $s_0, s_1, \ldots , s_{n-2}$.

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theorem Equiv.Perm.generated_by_simples {n : ℕ} :

Subgroup.closure {σ : Perm (Fin n) | ∃ (k : Fin (n - 1)), σ = simpleTransposition k} = ⊤

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Proposition 3.107

Let $n \in \mathbb {N}$ and $\sigma \in S_n$. For each $i \in [n]$, set

\[ a_i := s_{i'-1} s_{i'-2} \cdots s_i, \]

where $i' = i + \ell _i(\sigma )$. Then $\sigma = a_0 a_1 \cdots a_{n-1}$.

In other words, the product of the canonical reduced word equals $\sigma $.

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theorem Equiv.Perm.canonicalReducedWord_prod {n : ℕ} (σ : Perm (Fin n)) :

wordProd σ.canonicalReducedWord = σ

Signs of permutations

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Definition 3.109

Let $n \in \mathbb {N}$. The sign of a permutation $\sigma \in S_n$ is defined to be the integer $(-1)^{\ell (\sigma )}$.

It is denoted by $(-1)^{\sigma }$ or $\operatorname {sgn}(\sigma )$ or $\operatorname {sign}(\sigma )$ or $\varepsilon (\sigma )$. It is also known as the signature of $\sigma $.

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theorem Equiv.Perm.sign_eq_neg_one_pow_invCount {n : ℕ} (σ : Perm (Fin n)) :

sign σ = (-1) ^ AlgebraicCombinatorics.Perm.invCount σ

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theorem Equiv.Perm.sign_coe_eq_neg_one_pow_invCount {n : ℕ} (σ : Perm (Fin n)) :

↑(sign σ) = (-1) ^ AlgebraicCombinatorics.Perm.invCount σ

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Proposition 3.110

Let $n \in \mathbb {N}$.

(a) The sign of the identity permutation $\operatorname {id} \in S_n$ is $(-1)^{\operatorname {id}} = 1$.

(b) For any two distinct elements $i$ and $j$ of $[n]$, the transposition $t_{i,j} \in S_n$ has sign $(-1)^{t_{i,j}} = -1$.

(c) For any positive integer $k$ and any distinct elements $i_1, i_2, \ldots , i_k \in [n]$, the $k$-cycle $\operatorname {cyc}_{i_1, i_2, \ldots , i_k}$ has sign $(-1)^{\operatorname {cyc}_{i_1, i_2, \ldots , i_k}} = (-1)^{k-1}$.

(d) We have $(-1)^{\sigma \tau } = (-1)^{\sigma } \cdot (-1)^{\tau }$ for any $\sigma \in S_n$ and $\tau \in S_n$.

(e) We have $(-1)^{\sigma _1 \sigma _2 \cdots \sigma _p} = (-1)^{\sigma _1} (-1)^{\sigma _2} \cdots (-1)^{\sigma _p}$ for any $\sigma _1, \sigma _2, \ldots , \sigma _p \in S_n$.

(f) We have $(-1)^{\sigma ^{-1}} = (-1)^{\sigma }$ for any $\sigma \in S_n$. (The left hand side here has to be understood as $(-1)^{(\sigma ^{-1})}$.)

(g) We have

\[ (-1)^{\sigma } = \prod _{1 \leq i {\lt} j \leq n} \frac{\sigma (i) - \sigma (j)}{i - j} \qquad \text{for each } \sigma \in S_n. \]

(The product sign “$\prod _{1 \leq i {\lt} j \leq n}$” means a product over all pairs $(i,j)$ of integers satisfying $1 \leq i {\lt} j \leq n$. There are $\binom {n}{2}$ such pairs.)

(h) If $x_1, x_2, \ldots , x_n$ are any elements of some commutative ring, and if $\sigma \in S_n$, then

\[ \prod _{1 \leq i {\lt} j \leq n} \left( x_{\sigma (i)} - x_{\sigma (j)} \right) = (-1)^{\sigma } \prod _{1 \leq i {\lt} j \leq n} \left( x_i - x_j \right). \]

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theorem Equiv.Perm.sign_id {α : Type u_1} [DecidableEq α] [Fintype α] :

sign 1 = 1

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theorem Equiv.Perm.sign_transposition {α : Type u_1} [DecidableEq α] [Fintype α] {x y : α} (hxy : x ≠ y) :

sign (swap x y) = -1

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theorem Equiv.Perm.sign_isCycle {α : Type u_1} [DecidableEq α] [Fintype α] {σ : Perm α} (hσ : σ.IsCycle) :

sign σ = -(-1) ^ σ.support.card

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theorem Equiv.Perm.sign_mul' {α : Type u_1} [DecidableEq α] [Fintype α] (σ τ : Perm α) :

sign (σ * τ) = sign σ * sign τ

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theorem Equiv.Perm.sign_prod_list {α : Type u_1} [DecidableEq α] [Fintype α] (l : List (Perm α)) :

sign l.prod = (List.map (⇑sign) l).prod

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theorem Equiv.Perm.sign_inv' {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Perm α) :

sign σ⁻¹ = sign σ

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theorem Equiv.Perm.sign_eq_prod_pairs {n : ℕ} (σ : Perm (Fin n)) :

sign σ = ∏ i : Fin n, ∏ j ∈ Finset.Ioi i, if σ i < σ j then 1 else -1

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theorem Equiv.Perm.prod_diff_comp_perm {n : ℕ} {R : Type u_2} [CommRing R] (σ : Perm (Fin n)) (x : Fin n → R) :

∏ i : Fin n, ∏ j ∈ Finset.Ioi i, (x (σ i) - x (σ j)) = ↑↑(sign σ) * ∏ i : Fin n, ∏ j ∈ Finset.Ioi i, (x i - x j)

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Corollary 3.111

Let $n \in \mathbb {N}$. The map

\begin{align*} S_n & \to \{ 1, -1\} , \\ \sigma & \mapsto (-1)^{\sigma } \end{align*}

is a group homomorphism from the symmetric group $S_n$ to the order-$2$ group $\{ 1, -1\} $. (Of course, $\{ 1, -1\} $ is a group with respect to multiplication.)

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theorem Equiv.Perm.sign_hom_mul {α : Type u_1} [DecidableEq α] [Fintype α] (σ τ : Perm α) :

sign (σ * τ) = sign σ * sign τ

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Definition 3.112

Let $n \in \mathbb {N}$. A permutation $\sigma \in S_n$ is said to be

even if $(-1)^{\sigma } = 1$ (that is, if $\ell (\sigma )$ is even);
odd if $(-1)^{\sigma } = -1$ (that is, if $\ell (\sigma )$ is odd).

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def Equiv.Perm.IsEven {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Perm α) :

σ.IsEven = (Equiv.Perm.sign σ = 1)

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def Equiv.Perm.IsOdd {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Perm α) :

σ.IsOdd = (Equiv.Perm.sign σ = -1)

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Corollary 3.119

Let $n \in \mathbb {N}$. The set of all even permutations in $S_n$ is a normal subgroup of $S_n$.

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theorem Equiv.Perm.alternatingGroup_isNormal {α : Type u_1} [DecidableEq α] [Fintype α] :

(alternatingGroup α).Normal

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Corollary 3.120

Let $n \geq 2$. Then,

\[ \left(\text{\# of even permutations } \sigma \in S_n \right) = \left(\text{\# of odd permutations } \sigma \in S_n \right) = n!/2. \]

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theorem Equiv.Perm.card_even_perms {α : Type u_1} [DecidableEq α] [Fintype α] [Nontrivial α] :

Fintype.card ↥(alternatingGroup α) = (Fintype.card α).factorial / 2

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theorem Equiv.Perm.card_odd_eq_card_even {α : Type u_1} [DecidableEq α] [Fintype α] [Nontrivial α] :

{σ : Perm α | sign σ = -1}.card = {σ : Perm α | sign σ = 1}.card

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Proposition 3.122

Let $X$ be a finite set. We want to define the sign of any permutation of $X$.

Fix a bijection $\phi : X \to [n]$ for some $n \in \mathbb {N}$. (Such a bijection always exists, since $X$ is finite.) For every permutation $\sigma $ of $X$, set

\[ (-1)_{\phi }^{\sigma } := (-1)^{\phi \circ \sigma \circ \phi ^{-1}}. \]

Here, the right hand side is well-defined, since $\phi \circ \sigma \circ \phi ^{-1}$ is a permutation of $[n]$. Now:

(a) This number $(-1)_{\phi }^{\sigma }$ depends only on the permutation $\sigma $, but not on the bijection $\phi $. (In other words, if $\phi _1$ and $\phi _2$ are two bijections from $X$ to $[n]$, then $(-1)_{\phi _1}^{\sigma } = (-1)_{\phi _2}^{\sigma }$.)

Thus, we shall denote $(-1)_{\phi }^{\sigma }$ by $(-1)^{\sigma }$ from now on. We refer to this number $(-1)^{\sigma }$ as the sign of the permutation $\sigma \in S_X$. (When $X = [n]$, this notation does not clash with Definition 3.109, since we can pick the bijection $\phi = \operatorname {id}$ and obtain $(-1)_{\phi }^{\sigma } = (-1)^{\operatorname {id} \circ \sigma \circ \operatorname {id}^{-1}} = (-1)^{\sigma }$.)

(b) The identity permutation $\operatorname {id} : X \to X$ satisfies $(-1)^{\operatorname {id}} = 1$.

(c) We have $(-1)^{\sigma \tau } = (-1)^{\sigma } \cdot (-1)^{\tau }$ for any two permutations $\sigma $ and $\tau $ of $X$.

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theorem Equiv.Perm.sign_conj_eq {α : Type u_1} [DecidableEq α] [Fintype α] {β : Type u_2} [DecidableEq β] [Fintype β] (σ : Perm α) (e : α ≃ β) :

sign ((e.symm.trans σ).trans e) = sign σ

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theorem Equiv.Perm.sign_id_finiteSet {α : Type u_1} [DecidableEq α] [Fintype α] :

sign 1 = 1

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theorem Equiv.Perm.sign_mul_finiteSet {α : Type u_1} [DecidableEq α] [Fintype α] (σ τ : Perm α) :

sign (σ * τ) = sign σ * sign τ

The cycle decomposition

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Theorem 3.123 disjoint cycle decomposition of permutations

Let $X$ be a finite set. Let $\sigma $ be a permutation of $X$. Then:

(a) There is a list

\begin{align*} & \Big(\left( a_{1,1},a_{1,2},\ldots ,a_{1,n_{1}}\right) ,\\ & \ \ \ \left( a_{2,1},a_{2,2},\ldots ,a_{2,n_{2}}\right) ,\\ & \ \ \ \ldots ,\\ & \ \ \ \left( a_{k,1},a_{k,2},\ldots ,a_{k,n_{k}}\right) \Big) \end{align*}

of nonempty lists of elements of $X$ such that:

each element of $X$ appears exactly once in the composite list

\begin{align*} & (a_{1,1},a_{1,2},\ldots ,a_{1,n_{1}},\\ & \ \ \ a_{2,1},a_{2,2},\ldots ,a_{2,n_{2}},\\ & \ \ \ \ldots ,\\ & \ \ \ a_{k,1},a_{k,2},\ldots ,a_{k,n_{k}}), \end{align*}

and
we have

\[ \sigma =\operatorname *{cyc}\nolimits _{a_{1,1},a_{1,2},\ldots ,a_{1,n_{1}}}\circ \operatorname *{cyc}\nolimits _{a_{2,1},a_{2,2},\ldots ,a_{2,n_{2}}}\circ \cdots \circ \operatorname *{cyc}\nolimits _{a_{k,1},a_{k,2},\ldots ,a_{k,n_{k}}}. \]

Such a list is called a disjoint cycle decomposition (or short DCD) of $\sigma $. Its entries (which themselves are lists of elements of $X$) are called the cycles of $\sigma $.

(b) Any two DCDs of $\sigma $ can be obtained from each other by (repeatedly) swapping the cycles with each other, and rotating each cycle (i.e., replacing $\left( a_{i,1},a_{i,2},\ldots ,a_{i,n_{i}}\right) $ by $\left( a_{i,2},a_{i,3},\ldots ,a_{i,n_{i}},a_{i,1}\right) $).

(c) Now assume that $X$ is a set of integers (or, more generally, any totally ordered finite set). Then, there is a unique DCD

of $\sigma $ that satisfies the additional requirements that

we have $a_{i,1}\leq a_{i,p}$ for each $i\in \left[ k\right] $ and each $p\in \left[ n_{i}\right] $ (that is, each cycle in this DCD is written with its smallest entry first), and
we have $a_{1,1}{\gt}a_{2,1}{\gt}\cdots {\gt}a_{k,1}$ (that is, the cycles appear in this DCD in the order of decreasing first entries).

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theorem AlgebraicCombinatorics.CycleDecomposition.exists_dcd {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Equiv.Perm α) :

∃ (cycs : Finset (Equiv.Perm α)), (∀ c ∈ cycs, c.IsCycle) ∧ (↑cycs).Pairwise Equiv.Perm.Disjoint ∧ ∃ (h : (↑cycs).Pairwise Commute), cycs.noncommProd id h = σ

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theorem AlgebraicCombinatorics.CycleDecomposition.dcd_unique_cycleType {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Equiv.Perm α) (cycs : Finset (Equiv.Perm α)) :

(∀ c ∈ cycs, c.IsCycle) → (↑cycs).Pairwise Equiv.Perm.Disjoint → (∃ (h : (↑cycs).Pairwise Commute), cycs.noncommProd id h = σ) → Multiset.map (fun (c : Equiv.Perm α) => c.support.card) cycs.val = σ.cycleType

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theorem AlgebraicCombinatorics.CycleDecomposition.canonicalDcd_exists_unique {α : Type u_1} [DecidableEq α] [Fintype α] [LinearOrder α] [Inhabited α] (σ : Equiv.Perm α) :

∃! cycleReps : List (List α), (∀ l ∈ cycleReps, l ≠ []) ∧ cycleReps.flatten.Nodup ∧ (∀ (x : α), x ∈ cycleReps.flatten) ∧ (∀ l ∈ cycleReps, ∀ x ∈ l, l.head! ≤ x) ∧ List.Pairwise (fun (l1 l2 : List α) => l1.head! > l2.head!) cycleReps ∧ dcdListToPerm cycleReps = σ

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Definition 3.124

Let $X$ be a finite set. Let $\sigma $ be a permutation of $X$.

(a) The cycles of $\sigma $ are defined to be the cycles in the DCD of $\sigma $ (as defined in Theorem 3.123 (a)). (This includes $1$-cycles, if there are any in the DCD of $\sigma $.)

We shall equate a cycle of $\sigma $ with any of its cyclic rotations; thus, for example, $\left( 3,1,4\right) $ and $\left( 1,4,3\right) $ shall be regarded as being the same cycle (but $\left( 3,1,4\right) $ and $\left( 3,4,1\right) $ shall not).

(b) The cycle lengths partition of $\sigma $ shall denote the partition of $\left\vert X\right\vert $ obtained by writing down the lengths of the cycles of $\sigma $ in weakly decreasing order.

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def AlgebraicCombinatorics.CycleDecomposition.cycles {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Equiv.Perm α) :

Finset (Equiv.Perm α)

AlgebraicCombinatorics.CycleDecomposition.cycles σ = σ.cycleFactorsFinset

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def AlgebraicCombinatorics.CycleDecomposition.cycleLengthsPartition {α : Type u_1} [DecidableEq α] [Fintype α] (σ : Equiv.Perm α) :

(Fintype.card α).Partition

AlgebraicCombinatorics.CycleDecomposition.cycleLengthsPartition σ = σ.partition

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Proposition 3.135

Let $X$ be a finite set. Let $i,j\in X$. Let $\sigma $ be a permutation of $X$. Then, we have the following chain of equivalences:

\begin{align*} \ & \left( i\text{ and }j\text{ belong to the same cycle of }\sigma \right) \\ \Longleftrightarrow \ & \left( i=\sigma ^{p}\left( j\right) \text{ for some }p\in \mathbb {N}\right) \\ \Longleftrightarrow \ & \left( j=\sigma ^{p}\left( i\right) \text{ for some }p\in \mathbb {N}\right) . \end{align*}

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theorem AlgebraicCombinatorics.CycleDecomposition.sameCycle_iff_exists_pow {α : Type u_1} [Fintype α] (f : Equiv.Perm α) (i j : α) :

f.SameCycle i j ↔ ∃ (p : ℕ), (f ^ p) i = j

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Proposition 3.139

Let $n\in \mathbb {N}$. Let $\sigma \in S_{n}$. Let $k\in \mathbb {N}$ be such that $\sigma $ has exactly $k$ cycles (including the $1$-cycles). Then, $\left( -1\right) ^{\sigma }=\left( -1\right) ^{n-k}$.

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theorem AlgebraicCombinatorics.CycleDecomposition.sign_eq_neg_one_pow_card_sub_numCycles {n : ℕ} (σ : Equiv.Perm (Fin n)) :

Equiv.Perm.sign σ = (-1) ^ (n - numCyclesTotal σ)

Cancellations in alternating sums

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Proposition 4.6 Negative hockey-stick identity

Let $n\in \mathbb {C}$ and $m\in \mathbb {N}$. Then,

\[ \sum _{k=0}^{m} (-1)^{k} \binom {n}{k} = (-1)^{m} \binom {n-1}{m}. \]

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theorem AlgebraicCombinatorics.SignedCounting.negHockeyStick (n m : ℕ) (hn : 0 < n) :

∑ k ∈ Finset.range (m + 1), (-1) ^ k * ↑(n.choose k) = (-1) ^ m * ↑((n - 1).choose m)

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Lemma 4.9 Cancellation principle, take 1

Let $\mathcal{A}$ be a finite set. Let $\mathcal{X}$ be a subset of $\mathcal{A}$.

For each $I \in \mathcal{A}$, let $\operatorname {sign} I$ be an element of an additive abelian group $R$ with no $2$-torsion (i.e., $2a = 0$ implies $a = 0$). Let $f : \mathcal{X} \to \mathcal{X}$ be a bijection with the property that

\[ \operatorname {sign}(f(I)) = -\operatorname {sign} I \qquad \text{for all } I \in \mathcal{X}. \]

(Such a bijection $f$ is called sign-reversing.) Then,

\[ \sum _{I \in \mathcal{A}} \operatorname {sign} I = \sum _{I \in \mathcal{A} \setminus \mathcal{X}} \operatorname {sign} I. \]

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theorem AlgebraicCombinatorics.SignedCounting.sign_cancel1 {α : Type u_1} [DecidableEq α] {R : Type u_2} [AddCommGroup R] [NoZeroSMulDivisors ℕ R] (A X : Finset α) (hXA : X ⊆ A) (sign : α → R) (f : ↥X → ↥X) (hf_bij : Function.Bijective f) (hf_sign : ∀ (I : ↥X), sign ↑(f I) = -sign ↑I) :

∑ I ∈ A, sign I = ∑ I ∈ A \ X, sign I

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Lemma 4.10 Cancellation principle, take 2

Let $\mathcal{A}$ be a finite set. Let $\mathcal{X}$ be a subset of $\mathcal{A}$.

For each $I \in \mathcal{A}$, let $\operatorname {sign} I$ be an element of some additive abelian group. Let $f : \mathcal{X} \to \mathcal{X}$ be an involution (i.e., a map satisfying $f \circ f = \operatorname {id}$) that has no fixed points. Assume that

\[ \operatorname {sign}(f(I)) = -\operatorname {sign} I \qquad \text{for all } I \in \mathcal{X}. \]

Then,

\[ \sum _{I \in \mathcal{A}} \operatorname {sign} I = \sum _{I \in \mathcal{A} \setminus \mathcal{X}} \operatorname {sign} I. \]

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theorem AlgebraicCombinatorics.SignedCounting.sign_cancel2 {α : Type u_1} [DecidableEq α] {R : Type u_2} [AddCommGroup R] (A X : Finset α) (hXA : X ⊆ A) (sign : α → R) (f : ↥X → ↥X) (hf_invol : ∀ (I : ↥X), f (f I) = I) (hf_no_fixed : ∀ (I : ↥X), f I ≠ I) (hf_sign : ∀ (I : ↥X), sign ↑(f I) = -sign ↑I) :

∑ I ∈ A, sign I = ∑ I ∈ A \ X, sign I

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Lemma 4.11 Cancellation principle, take 3

Let $\mathcal{A}$ be a finite set. Let $\mathcal{X}$ be a subset of $\mathcal{A}$.

\[ \operatorname {sign}(f(I)) = -\operatorname {sign} I \qquad \text{for all } I \in \mathcal{X}. \]

Assume furthermore that

\[ \operatorname {sign} I = 0 \qquad \text{for all } I \in \mathcal{X} \text{ satisfying } f(I) = I. \]

Then,

\[ \sum _{I \in \mathcal{A}} \operatorname {sign} I = \sum _{I \in \mathcal{A} \setminus \mathcal{X}} \operatorname {sign} I. \]

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theorem AlgebraicCombinatorics.SignedCounting.sign_cancel3 {α : Type u_1} [DecidableEq α] {R : Type u_2} [AddCommGroup R] (A X : Finset α) (hXA : X ⊆ A) (sign : α → R) (f : ↥X → ↥X) (hf_invol : ∀ (I : ↥X), f (f I) = I) (hf_sign : ∀ (I : ↥X), sign ↑(f I) = -sign ↑I) (hf_fixed_zero : ∀ (I : ↥X), f I = I → sign ↑I = 0) :

∑ I ∈ A, sign I = ∑ I ∈ A \ X, sign I

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Definition 4.15

Let $K$ be a field. Let $d$ be a positive integer.

(a) A $d$-th root of unity in $K$ means an element $\omega $ of $K$ satisfying $\omega ^d = 1$.

(b) A primitive $d$-th root of unity in $K$ means an element $\omega $ of $K$ satisfying $\omega ^d = 1$ but $\omega ^i \neq 1$ for each $i \in \{ 1, 2, \ldots , d-1\} $. In other words, a primitive $d$-th root of unity in $K$ means an element of the multiplicative group $K^{\times }$ whose order is $d$.

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def AlgebraicCombinatorics.SignedCounting.IsRootOfUnity {K : Type u_1} [CommRing K] (ω : K) (d : ℕ) :

AlgebraicCombinatorics.SignedCounting.IsRootOfUnity ω d = (ω ^ d = 1)

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structure IsPrimitiveRoot {M : Type u_1} [CommMonoid M] (ζ : M) (k : ℕ) :

pow_eq_one : ζ ^ k = 1
dvd_of_pow_eq_one (l : ℕ) : ζ ^ l = 1 → k ∣ l

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Theorem 4.28 $q$-Lucas theorem

Let $K$ be a field. Let $d$ be a positive integer. Let $\omega $ be a primitive $d$-th root of unity in $K$. Let $n, k \in \mathbb {N}$. Let $q$ and $u$ be the quotients obtained when dividing $n$ and $k$ by $d$ with remainder, and let $r$ and $v$ be the respective remainders. Then,

\[ \binom {n}{k}_{\omega } = \binom {q}{u} \cdot \binom {r}{v}_{\omega }. \]

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theorem AlgebraicCombinatorics.SignedCounting.qLucas {K : Type u_1} [Field K] (d : ℕ) (hd : 0 < d) (ω : K) (hω : IsPrimitiveRoot ω d) (n k : ℕ) :

(Polynomial.aeval ω) (qBinomial n k) = ↑((n / d).choose (k / d)) * (Polynomial.aeval ω) (qBinomial (n % d) (k % d))

The principles of inclusion and exclusion

The size version

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Theorem 4.43 Size version of the PIE

Let $n \in \mathbb {N}$. Let $U$ be a finite set. Let $A_1, A_2, \ldots , A_n$ be $n$ subsets of $U$. Then,

\[ \left(\text{\# of } u \in U \text{ that satisfy } u \notin A_i \text{ for all } i \in [n]\right) = \sum _{I \subseteq [n]} (-1)^{|I|} \left(\text{\# of } u \in U \text{ that satisfy } u \in A_i \text{ for all } i \in I\right). \]

Equivalently,

\[ \left| U \setminus (A_1 \cup A_2 \cup \cdots \cup A_n) \right| = \sum _{I \subseteq [n]} (-1)^{|I|} \left| \bigcap _{i \in I} A_i \right|, \]

where $\bigcap _{i \in I} A_i$ denotes $\{ u \in U \mid u \in A_i \text{ for all } i \in I\} $ (so that $\bigcap _{i \in \varnothing } A_i = U$).

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theorem AlgebraicCombinatorics.InclusionExclusion.pie_size_version {α : Type u_1} {ι : Type u_2} [DecidableEq α] [Fintype α] (s : Finset ι) (A : ι → Finset α) :

↑(s.inf fun (i : ι) => (A i)ᶜ).card = ∑ I ∈ s.powerset, (-1) ^ I.card * ↑(I.inf A).card

Examples

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Theorem 4.48

Let $n, m \in \mathbb {N}$. Then,

\[ \left(\text{\# of surjective maps } f\colon [m] \to [n]\right) = \sum _{k=0}^{n} (-1)^k \binom {n}{k} (n-k)^m. \]

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theorem AlgebraicCombinatorics.InclusionExclusion.numSurj_formula (m n : ℕ) :

↑(numSurj m n) = ∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑(n.choose k) * (↑n - ↑k) ^ m

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Corollary 4.51

Let $n \in \mathbb {N}$. Then:

(a) We have $\sum _{k=0}^{n} (-1)^k \binom {n}{k} (n-k)^m = 0$ for any $m \in \mathbb {N}$ satisfying $m {\lt} n$.

(b) We have $\sum _{k=0}^{n} (-1)^k \binom {n}{k} (n-k)^n = n!$.

(c) We have $\sum _{k=0}^{n} (-1)^k \binom {n}{k} (n-k)^m \geq 0$ for each $m \in \mathbb {N}$.

(d) We have $n! \mid \sum _{k=0}^{n} (-1)^k \binom {n}{k} (n-k)^m$ for each $m \in \mathbb {N}$.

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theorem AlgebraicCombinatorics.InclusionExclusion.surjOn_alternating_sum_eq_zero (n m : ℕ) (h : m < n) :

∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑(n.choose k) * (↑n - ↑k) ^ m = 0

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theorem AlgebraicCombinatorics.InclusionExclusion.surjOn_alternating_sum_eq_factorial (n : ℕ) :

∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑(n.choose k) * (↑n - ↑k) ^ n = ↑n.factorial

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theorem AlgebraicCombinatorics.InclusionExclusion.surjOn_alternating_sum_nonneg (m n : ℕ) :

0 ≤ ∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑(n.choose k) * (↑n - ↑k) ^ m

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theorem AlgebraicCombinatorics.InclusionExclusion.surjOn_alternating_sum_dvd_factorial (m n : ℕ) :

↑n.factorial ∣ ∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑(n.choose k) * (↑n - ↑k) ^ m

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Definition 4.56

A derangement of a set $X$ means a permutation of $X$ that has no fixed points.

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def AlgebraicCombinatorics.InclusionExclusion.Derangement (α : Type u_1) :

Type u_1

AlgebraicCombinatorics.InclusionExclusion.Derangement α = { σ : Equiv.Perm α // ∀ (x : α), σ x ≠ x }

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Theorem 4.62

Let $n \in \mathbb {N}$. Then, the number of derangements of $[n]$ is

\[ D_n = \sum _{k=0}^{n} (-1)^k \binom {n}{k} (n-k)! = n! \cdot \sum _{k=0}^{n} \frac{(-1)^k}{k!}. \]

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theorem AlgebraicCombinatorics.InclusionExclusion.numDerangements_formula (n : ℕ) :

↑(numDerangements n) = ∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑(n.choose k) * ↑(n - k).factorial

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Theorem 4.64

Let $c$ be a positive integer with prime factorization $p_1^{a_1} p_2^{a_2} \cdots p_n^{a_n}$, where $p_1, p_2, \ldots , p_n$ are distinct primes and $a_1, a_2, \ldots , a_n$ are positive integers. Then,

\[ \left(\text{\# of all } u \in [c] \text{ that are coprime to } c\right) = c \cdot \prod _{i=1}^{n} \left(1 - \frac{1}{p_i}\right) = \prod _{i=1}^{n} \left(p_i^{a_i} - p_i^{a_i - 1}\right). \]

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theorem AlgebraicCombinatorics.InclusionExclusion.totient_eq_prod_one_sub_inv (c : ℕ) (_hc : 0 < c) :

↑c.totient = ↑c * ∏ p ∈ c.primeFactors, (1 - 1 / ↑p)

The weighted version

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Theorem 4.65 Weighted version of the PIE

Let $n \in \mathbb {N}$, and let $U$ be a finite set. Let $A_1, A_2, \ldots , A_n$ be $n$ subsets of $U$. Let $G$ be any additive abelian group. Let $w\colon U \to G$ be any map. Then,

\[ \sum _{\substack {u \in U \\ u \notin A_i \text{ for all } i \in [n]}} w(u) = \sum _{I \subseteq [n]} (-1)^{|I|} \sum _{\substack {u \in U \\ u \in A_i \text{ for all } i \in I}} w(u). \]

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theorem AlgebraicCombinatorics.InclusionExclusion.weighted_pie {ι : Type u_1} {U : Type u_2} {G : Type u_3} [Fintype ι] [DecidableEq ι] [Fintype U] [DecidableEq U] [AddCommGroup G] (A : ι → Finset U) (w : U → G) :

∑ u ∈ Finset.univ.inf fun (i : ι) => (A i)ᶜ, w u = ∑ I ∈ Finset.univ.powerset, (-1) ^ I.card • ∑ u ∈ I.inf A, w u

Boolean Möbius inversion

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Theorem 4.70 Boolean Möbius inversion

Let $S$ be a finite set. Let $A$ be any additive abelian group.

For each subset $I$ of $S$, let $a_{I}$ and $b_{I}$ be two elements of $A$.

Assume that

\begin{equation} b_{I}=\sum _{J\subseteq I}a_{J}\ \ \ \ \ \ \ \ \ \ \text{for all }I\subseteq S. \label{eq.thm.pie.moeb.ass} \end{equation}

Then, we also have

\begin{equation} a_{I}=\sum _{J\subseteq I}\left( -1\right) ^{\left\vert I\setminus J\right\vert }b_{J}\ \ \ \ \ \ \ \ \ \ \text{for all }I\subseteq S. \label{eq.thm.pie.moeb.claim} \end{equation}

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theorem BooleanMobius.booleanMobiusInversion {α : Type u_1} [DecidableEq α] {S : Finset α} {A : Type u_2} [AddCommGroup A] (a b : Finset α → A) (hab : ∀ I ⊆ S, b I = ∑ J ∈ I.powerset, a J) (I : Finset α) :

I ⊆ S → a I = ∑ J ∈ I.powerset, (-1) ^ (I \ J).card • b J

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Lemma 4.68 Alternating sum over supersets

Let $P \subseteq Q$ be two finite sets.

(a) We have

\begin{equation} \sum _{\substack {I\subseteq Q;\\ P\subseteq I}}\left( -1\right) ^{\left\vert I\right\vert }=\left( -1\right) ^{\left\vert P\right\vert }\left[ P=Q\right] . \label{eq.lem.pie.two-sets-altsum.a} \end{equation}

(b) We have

\begin{equation} \sum _{\substack {I\subseteq Q;\\ P\subseteq I}}\left( -1\right) ^{\left\vert Q\setminus I\right\vert }=\left[ P=Q\right] . \label{eq.lem.pie.two-sets-altsum.b} \end{equation}

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theorem BooleanMobius.alternatingSum_superset_eq_iverson_a {α : Type u_1} [DecidableEq α] {P Q : Finset α} (hPQ : P ⊆ Q) :

∑ I ∈ Q.powerset with P ⊆ I, (-1) ^ I.card = (-1) ^ P.card * if P = Q then 1 else 0

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theorem BooleanMobius.alternatingSum_superset_eq_iverson_b {α : Type u_1} [DecidableEq α] {P Q : Finset α} (hPQ : P ⊆ Q) :

∑ I ∈ Q.powerset with P ⊆ I, (-1) ^ (Q \ I).card = if P = Q then 1 else 0

More subtractive methods

Sums with varying signs

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Theorem 4.78

Let $n\in \mathbb {N}$ and $d\in \mathbb {N}$. Then, the number of all all-even $n$-tuples $\left( x_{1},x_{2},\ldots ,x_{n}\right) \in \left[ d\right] ^{n}$ is

\[ \frac{1}{2^{d}}\sum _{k=0}^{d}\binom {d}{k}\left( d-2k\right) ^{n}. \]

More precisely (avoiding division), the number of all-even $n$-tuples multiplied by $2^d$ equals $\sum _{k=0}^{d}\binom {d}{k}(d-2k)^n$.

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theorem AlgebraicCombinatorics.SubtractiveMethods.allEven_count_formula (n d : ℕ) :

↑(allEvenTuples n d).card * 2 ^ d = ∑ k ∈ Finset.range (d + 1), ↑(d.choose k) * (↑d - 2 * ↑k) ^ n

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Lemma 4.80

Let $n,d\in \mathbb {N}$. Then,

\begin{align*} & \sum _{\left( e_{1},e_{2},\ldots ,e_{d}\right) \in \left\{ 1,-1\right\} ^{d}}\left( e_{1}+e_{2}+\cdots +e_{d}\right) ^{n}\\ & =\sum _{\left( x_{1},x_{2},\ldots ,x_{n}\right) \in \left[ d\right] ^{n}}\ \ \sum _{\left( e_{1},e_{2},\ldots ,e_{d}\right) \in \left\{ 1,-1\right\} ^{d}}e_{x_{1}}e_{x_{2}}\cdots e_{x_{n}}. \end{align*}

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theorem AlgebraicCombinatorics.SubtractiveMethods.sum_signSum_pow_eq_sum_signProduct (n d : ℕ) :

∑ e : Fin d → ZMod 2, signSum e ^ n = ∑ x : Fin n → Fin d, ∑ e : Fin d → ZMod 2, signProduct e x

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Lemma 4.81

Let $n,d\in \mathbb {N}$. Let $\left( x_{1},x_{2},\ldots ,x_{n}\right) \in \left[ d\right] ^{n}$.

(a) If the $n$-tuple $\left( x_{1},x_{2},\ldots ,x_{n}\right)$ is not all-even, then $\sum _{\left( e_{1},\ldots ,e_{d}\right) \in \left\{ 1,-1\right\} ^{d}} e_{x_{1}}e_{x_{2}}\cdots e_{x_{n}}=0$.

(b) If the $n$-tuple $\left( x_{1},x_{2},\ldots ,x_{n}\right)$ is all-even, then $\sum _{\left( e_{1},\ldots ,e_{d}\right) \in \left\{ 1,-1\right\} ^{d}} e_{x_{1}}e_{x_{2}}\cdots e_{x_{n}}=2^{d}$.

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theorem AlgebraicCombinatorics.SubtractiveMethods.sum_signProduct_not_allEven {n d : ℕ} (x : Fin n → Fin d) (hx : ¬IsAllEven x) :

∑ e : Fin d → ZMod 2, signProduct e x = 0

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theorem AlgebraicCombinatorics.SubtractiveMethods.sum_signProduct_allEven {n d : ℕ} (x : Fin n → Fin d) (hx : IsAllEven x) :

∑ e : Fin d → ZMod 2, signProduct e x = 2 ^ d

Determinants

Definition

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Definition 5.3

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix. The determinant $\det A$ of $A$ is defined to be the element

\[ \sum _{\sigma \in S_n} (-1)^{\sigma } \underbrace{A_{1,\sigma (1)} A_{2,\sigma (2)} \cdots A_{n,\sigma (n)}}_{ = \prod _{i=1}^{n} A_{i,\sigma (i)}} \]

of $K$. Here:

we let $S_n$ denote the $n$-th symmetric group (i.e., the group of permutations of $[n] = \{ 1, 2, \ldots , n\} $);
we let $(-1)^{\sigma }$ denote the sign of the permutation $\sigma $ (as defined in Definition 3.109).

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theorem AlgebraicCombinatorics.Det.det_eq_sum_sign_prod_textbook {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) :

A.det = ∑ σ : Equiv.Perm (Fin n), Equiv.Perm.sign σ • ∏ i : Fin n, A i (σ i)

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Proposition 5.12

Let $n \in \mathbb {N}$ be such that $n \geq 2$. Let $x_1, x_2, \ldots , x_n \in K$ and $y_1, y_2, \ldots , y_n \in K$. Then,

\[ \det \left( \left( x_i y_j \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = 0. \]

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theorem AlgebraicCombinatorics.Det.det_outerProduct_eq_zero {K : Type u_1} [CommRing K] {n : ℕ} (x y : Fin n → K) (hn : 2 ≤ n) :

(outerProductMatrix x y).det = 0

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Proposition 5.17

Let $n \in \mathbb {N}$ be such that $n \geq 3$. Let $x_1, x_2, \ldots , x_n \in K$ and $y_1, y_2, \ldots , y_n \in K$. Then,

\[ \det \left( \left( x_i + y_j \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = 0. \]

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theorem AlgebraicCombinatorics.Det.det_sumMatrix_eq_zero {K : Type u_1} [CommRing K] {n : ℕ} (x y : Fin n → K) (hn : 3 ≤ n) :

(sumMatrix x y).det = 0

Basic properties

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Theorem 5.18 Transposes preserve determinants

Let $n \in \mathbb {N}$. If $A \in K^{n \times n}$ is any $n \times n$-matrix, then $\det (A^T) = \det A$.

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theorem AlgebraicCombinatorics.Det.det_transpose' {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) :

A.transpose.det = A.det

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Theorem 5.19 Determinants of triangular matrices

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be a triangular (i.e., lower-triangular or upper-triangular) $n \times n$-matrix. Then, the determinant of the matrix $A$ is the product of its diagonal entries. That is,

\[ \det A = A_{1,1} A_{2,2} \cdots A_{n,n}. \]

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theorem AlgebraicCombinatorics.Det.det_upperTriangular {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (hA : ∀ (i j : Fin n), j < i → A i j = 0) :

A.det = ∏ i : Fin n, A i i

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theorem AlgebraicCombinatorics.Det.det_lowerTriangular {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (hA : ∀ (i j : Fin n), i < j → A i j = 0) :

A.det = ∏ i : Fin n, A i i

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Theorem 5.20 Row operation properties

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix. Then:

(a) If we swap two rows of $A$, then $\det A$ gets multiplied by $-1$.

(b) If $A$ has a zero row (i.e., a row that consists entirely of zeroes), then $\det A = 0$.

(c) If $A$ has two equal rows, then $\det A = 0$.

(d) Let $\lambda \in K$. If we multiply a row of $A$ by $\lambda $ (that is, we multiply all entries of this one row by $\lambda $, while leaving all other entries of $A$ unchanged), then $\det A$ gets multiplied by $\lambda $.

(e) If we add a row of $A$ to another row of $A$ (that is, we add each entry of the former row to the corresponding entry of the latter), then $\det A$ stays unchanged.

(f) Let $\lambda \in K$. If we add $\lambda $ times a row of $A$ to another row of $A$ (that is, we add $\lambda $ times each entry of the former row to the corresponding entry of the latter), then $\det A$ stays unchanged.

(g) Let $B, C \in K^{n \times n}$ be two further $n \times n$-matrices. Let $k \in [n]$. Assume that

\[ (\text{the } k\text{-th row of } C) = (\text{the } k\text{-th row of } A) + (\text{the } k\text{-th row of } B), \]

whereas each $i \neq k$ satisfies

\[ (\text{the } i\text{-th row of } C) = (\text{the } i\text{-th row of } A) = (\text{the } i\text{-th row of } B). \]

Then,

\[ \det C = \det A + \det B. \]

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theorem AlgebraicCombinatorics.Det.det_swap_rows {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) :

(Matrix.of (A ∘ ⇑(Equiv.swap i j))).det = -A.det

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theorem AlgebraicCombinatorics.Det.det_zero_row {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i : Fin n) (hi : ∀ (j : Fin n), A i j = 0) :

A.det = 0

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theorem AlgebraicCombinatorics.Det.det_eq_rows {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) (hrows : A i = A j) :

A.det = 0

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theorem AlgebraicCombinatorics.Det.det_scale_row {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i : Fin n) (c : K) :

(A.updateRow i (c • A i)).det = c * A.det

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theorem AlgebraicCombinatorics.Det.det_add_row {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) :

(A.updateRow i (A i + A j)).det = A.det

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theorem AlgebraicCombinatorics.Det.det_add_smul_row {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) (c : K) :

(A.updateRow i (A i + c • A j)).det = A.det

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theorem AlgebraicCombinatorics.Det.det_row_add {K : Type u_1} [CommRing K] {n : ℕ} (A B C : Matrix (Fin n) (Fin n) K) (k : Fin n) (hk : C k = A k + B k) (hother : ∀ (i : Fin n), i ≠ k → C i = A i ∧ A i = B i) :

C.det = A.det + B.det

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Theorem 5.21 Column operation properties

Theorem 5.20 also holds if we replace “row” by “column” throughout it.

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theorem AlgebraicCombinatorics.Det.det_swap_cols {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) :

(A.submatrix id ⇑(Equiv.swap i j)).det = -A.det

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theorem AlgebraicCombinatorics.Det.det_zero_col {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (j : Fin n) (hj : ∀ (i : Fin n), A i j = 0) :

A.det = 0

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theorem AlgebraicCombinatorics.Det.det_eq_cols {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) (hcols : ∀ (k : Fin n), A k i = A k j) :

A.det = 0

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theorem AlgebraicCombinatorics.Det.det_scale_col {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (j : Fin n) (c : K) :

(Matrix.of fun (i k : Fin n) => if k = j then c * A i k else A i k).det = c * A.det

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theorem AlgebraicCombinatorics.Det.det_add_col {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) :

(A.updateCol i fun (k : Fin n) => A k i + A k j).det = A.det

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theorem AlgebraicCombinatorics.Det.det_add_smul_col {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (i j : Fin n) (hij : i ≠ j) (c : K) :

(A.updateCol i fun (k : Fin n) => A k i + c * A k j).det = A.det

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theorem AlgebraicCombinatorics.Det.det_col_add {K : Type u_1} [CommRing K] {n : ℕ} (A B C : Matrix (Fin n) (Fin n) K) (k : Fin n) (hk : ∀ (i : Fin n), C i k = A i k + B i k) (hother : ∀ (i j : Fin n), j ≠ k → C i j = A i j ∧ A i j = B i j) :

C.det = A.det + B.det

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Corollary 5.22

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ and $\tau \in S_n$. Then,

\begin{equation} \det \left( \left( A_{\tau (i), j} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = (-1)^{\tau } \cdot \det A \label{eq.cor.det.sig-row-col.row} \end{equation}

and

\begin{equation} \det \left( \left( A_{i, \tau (j)} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = (-1)^{\tau } \cdot \det A. \label{eq.cor.det.sig-row-col.col} \end{equation}

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theorem AlgebraicCombinatorics.Det.det_permute_rows {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (τ : Equiv.Perm (Fin n)) :

(A.submatrix (⇑τ) id).det = ↑↑(Equiv.Perm.sign τ) * A.det

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theorem AlgebraicCombinatorics.Det.det_permute_cols {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (τ : Equiv.Perm (Fin n)) :

(A.submatrix id ⇑τ).det = ↑↑(Equiv.Perm.sign τ) * A.det

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Theorem 5.23 Multiplicativity of the determinant

Let $n \in \mathbb {N}$. Let $A, B \in K^{n \times n}$ be two $n \times n$-matrices. Then,

\[ \det (AB) = \det A \cdot \det B. \]

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theorem AlgebraicCombinatorics.Det.det_mul' {K : Type u_1} [CommRing K] {n : ℕ} (A B : Matrix (Fin n) (Fin n) K) :

(A * B).det = A.det * B.det

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Corollary 5.24

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ and $d_1, d_2, \ldots , d_n \in K$. Then,

\begin{equation} \det \left( \left( d_i A_{i,j} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = d_1 d_2 \cdots d_n \cdot \det A \label{eq.cor.det.scale-row-col.row} \end{equation}

and

\begin{equation} \det \left( \left( d_j A_{i,j} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = d_1 d_2 \cdots d_n \cdot \det A. \label{eq.cor.det.scale-row-col.col} \end{equation}

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theorem AlgebraicCombinatorics.Det.det_scale_rows {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (d : Fin n → K) :

(Matrix.of fun (i j : Fin n) => d i * A i j).det = (∏ i : Fin n, d i) * A.det

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theorem AlgebraicCombinatorics.Det.det_scale_cols {K : Type u_1} [CommRing K] {n : ℕ} (A : Matrix (Fin n) (Fin n) K) (d : Fin n → K) :

(Matrix.of fun (i j : Fin n) => d j * A i j).det = (∏ j : Fin n, d j) * A.det

Cauchy–Binet

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Theorem 5.39 Cauchy–Binet formula

Let $n,m\in \mathbb {N}$. Let $A\in K^{n\times m}$ be an $n\times m$-matrix, and let $B\in K^{m\times n}$ be an $m\times n$-matrix. Then,

\[ \det (AB) = \sum _{\substack {(g_1,g_2,\ldots ,g_n)\in [m]^n;\\ g_1{\lt}g_2{\lt}\cdots {\lt}g_n}} \det (\operatorname {cols}_{g_1,g_2,\ldots ,g_n} A) \cdot \det (\operatorname {rows}_{g_1,g_2,\ldots ,g_n} B). \]

Here, $\operatorname {cols}_{g_1,\ldots ,g_n} A$ is the $n\times n$-matrix obtained from $A$ by keeping only columns $g_1,g_2,\ldots ,g_n$, and $\operatorname {rows}_{g_1,\ldots ,g_n} B$ is the $n\times n$-matrix obtained from $B$ by keeping only rows $g_1,g_2,\ldots ,g_n$.

Equivalently, the sum runs over all $n$-element subsets $S$ of $[m]$:

\[ \det (AB) = \sum _{\substack {S \subseteq [m] \\ |S|=n}} \det (\operatorname {cols}_S A) \cdot \det (\operatorname {rows}_S B). \]

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theorem AlgebraicCombinatorics.CauchyBinet.cauchyBinet {R : Type u_1} [CommRing R] {n m : ℕ} (A : Matrix (Fin n) (Fin m) R) (B : Matrix (Fin m) (Fin n) R) :

(A * B).det = ∑ S ∈ Finset.powersetCard n Finset.univ, if h : S.card = n then (colsSubmatrix A S h).det * (rowsSubmatrix B S h).det else 0

$\det\left( A+B\right) $

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Definition 5.35

Let $n,m\in \mathbb {N}$. Let $A$ be an $n\times m$-matrix.

Let $U$ be a subset of $[n]$. Let $V$ be a subset of $[m]$.

Writing the two sets $U$ and $V$ as

\[ U=\{ u_1,u_2,\ldots ,u_p\} \quad \text{and}\quad V=\{ v_1,v_2,\ldots ,v_q\} \]

with

\[ u_1{\lt}u_2{\lt}\cdots {\lt}u_p \quad \text{and}\quad v_1{\lt}v_2{\lt}\cdots {\lt}v_q, \]

we set

\[ \operatorname {sub}_U^V A:=\left(A_{u_i,v_j}\right)_{1\le i\le p,\ 1\le j\le q}. \]

Roughly speaking, $\operatorname {sub}_U^V A$ is the matrix obtained from $A$ by focusing only on the $i$-th rows for $i\in U$ (that is, removing all the other rows) and only on the $j$-th columns for $j\in V$ (that is, removing all the other columns).

This matrix $\operatorname {sub}_U^V A$ is called the submatrix of $A$ obtained by restricting to the $U$-rows and the $V$-columns. If this matrix is square (i.e., if $|U|=|V|$), then its determinant $\det (\operatorname {sub}_U^V A)$ is called a minor of $A$.

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noncomputable def AlgebraicCombinatorics.CauchyBinet.submatrixOfFinset {R : Type u_1} {n m : ℕ} (A : Matrix (Fin n) (Fin m) R) (U : Finset (Fin n)) (V : Finset (Fin m)) :

Matrix (Fin U.card) (Fin V.card) R

AlgebraicCombinatorics.CauchyBinet.submatrixOfFinset A U V = A.submatrix ⇑(U.orderEmbOfFin ⋯) ⇑(V.orderEmbOfFin ⋯)

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Theorem 5.61

Let $n\in \mathbb {N}$. For any subset $I$ of $[n]$, let $\widetilde{I} = [n]\setminus I$ denote its complement.

Let $A$ and $B$ be two $n\times n$-matrices in $K^{n\times n}$. Then,

\[ \det (A+B) = \sum _{P\subseteq [n]}\; \; \sum _{\substack {Q\subseteq [n];\\ |P|=|Q|}} (-1)^{\operatorname {sum} P + \operatorname {sum} Q} \det (\operatorname {sub}_P^Q A) \cdot \det (\operatorname {sub}_{\widetilde{P}}^{\widetilde{Q}} B). \]

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theorem AlgebraicCombinatorics.CauchyBinet.det_add_sum {R : Type u_1} [CommRing R] {n : ℕ} (A B : Matrix (Fin n) (Fin n) R) :

(A + B).det = ∑ P : Finset (Fin n), ∑ Q : Finset (Fin n), if h : Q.card = P.card then (-1) ^ (finsetSumFin P + finsetSumFin Q) * submatrixDet A Q P h * submatrixDet B Qᶜ Pᶜ ⋯ else 0

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Lemma 5.62

Let $n\in \mathbb {N}$. Let $d_1,d_2,\ldots ,d_n\in K$. Let

\[ D:=\operatorname {diag}(d_1,d_2,\ldots ,d_n) =\begin{pmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{pmatrix}\in K^{n\times n}. \]

(a) We have $\det (\operatorname {sub}_P^P D) = \prod _{i\in P} d_i$ for any subset $P$ of $[n]$.

(b) Let $P$ and $Q$ be two distinct subsets of $[n]$ satisfying $|P|=|Q|$. Then, $\det (\operatorname {sub}_P^Q D) = 0$.

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theorem AlgebraicCombinatorics.CauchyBinet.det_diagonal_submatrix_eq {R : Type u_1} [CommRing R] {n : ℕ} (d : Fin n → R) (P : Finset (Fin n)) :

((Matrix.diagonal d).submatrix ⇑(P.orderEmbOfFin ⋯) ⇑(P.orderEmbOfFin ⋯)).det = ∏ i ∈ P, d i

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theorem AlgebraicCombinatorics.CauchyBinet.det_diagonal_submatrix_off_diag {R : Type u_1} [CommRing R] {n : ℕ} (d : Fin n → R) (P Q : Finset (Fin n)) (hcard : P.card = Q.card) (hne : P ≠ Q) :

((Matrix.diagonal d).submatrix ⇑(P.orderEmbOfFin ⋯) ⇑(Q.orderEmbOfFin ⋯)).det = 0

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Theorem 5.63

Let $n\in \mathbb {N}$. Let $A$ and $D$ be two $n\times n$-matrices in $K^{n\times n}$ such that the matrix $D$ is diagonal. Let $d_1,d_2,\ldots ,d_n$ be the diagonal entries of the diagonal matrix $D$. Then,

\[ \det (A+D) = \sum _{P\subseteq [n]} \det (\operatorname {sub}_P^P A) \cdot \prod _{i\in [n]\setminus P} d_i. \]

The minors $\det (\operatorname {sub}_P^P A)$ are called the principal minors of $A$.

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theorem AlgebraicCombinatorics.CauchyBinet.det_add_diagonal {R : Type u_1} [CommRing R] {n : ℕ} (A : Matrix (Fin n) (Fin n) R) (d : Fin n → R) :

(A + Matrix.diagonal d).det = ∑ P : Finset (Fin n), (A.submatrix ⇑(P.orderEmbOfFin ⋯) ⇑(P.orderEmbOfFin ⋯)).det * ∏ i ∈ Pᶜ, d i

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Proposition 5.67

Let $n\in \mathbb {N}$. Let $d_1,d_2,\ldots ,d_n\in K$ and $x\in K$. Let $F$ be the $n\times n$-matrix

\[ \begin{pmatrix} x+d_1 & x & \cdots & x \\ x & x+d_2 & \cdots & x \\ \vdots & \vdots & \ddots & \vdots \\ x & x & \cdots & x+d_n \end{pmatrix}\in K^{n\times n}. \]

Then,

\[ \det F = d_1 d_2 \cdots d_n + x \sum _{i=1}^n d_1 d_2 \cdots \widehat{d_i} \cdots d_n, \]

where the hat over “$d_i$” means “omit the $d_i$ factor.”

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theorem AlgebraicCombinatorics.CauchyBinet.det_const_add_diagonal {R : Type u_1} [CommRing R] {n : ℕ} (x : R) (d : Fin n → R) :

(constPlusDiagMatrix x d).det = ∏ i : Fin n, d i + x * ∑ i : Fin n, ∏ j ∈ Finset.univ.erase i, d j

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Proposition 5.68

Let $n\in \mathbb {N}$. Let $A\in K^{n\times n}$ be an $n\times n$-matrix. Let $x\in K$. Let $I_n$ denote the $n\times n$ identity matrix. Then,

\[ \det (A + x I_n) = \sum _{P\subseteq [n]} \det (\operatorname {sub}_P^P A) \cdot x^{n-|P|} = \sum _{k=0}^{n} \biggl(\sum _{\substack {P\subseteq [n];\\ |P|=n-k}} \det (\operatorname {sub}_P^P A)\biggr) x^k. \]

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theorem AlgebraicCombinatorics.CauchyBinet.det_charPoly_coeff {R : Type u_1} [CommRing R] {n : ℕ} (A : Matrix (Fin n) (Fin n) R) (x : R) :

(A + x • 1).det = ∑ P : Finset (Fin n), (A.submatrix ⇑(P.orderEmbOfFin ⋯) ⇑(P.orderEmbOfFin ⋯)).det * x ^ (n - P.card)

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theorem AlgebraicCombinatorics.CauchyBinet.det_charPoly_coeff' {R : Type u_1} [CommRing R] {n : ℕ} (A : Matrix (Fin n) (Fin n) R) (x : R) :

(A + x • 1).det = ∑ k ∈ Finset.range (n + 1), (∑ P : Finset (Fin n) with P.card = n - k, (A.submatrix ⇑(P.orderEmbOfFin ⋯) ⇑(P.orderEmbOfFin ⋯)).det) * x ^ k

Factoring the matrix

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Proposition 5.74

Let $n\in \mathbb {N}$. Let $A$ be the $n\times n$-matrix

\[ \Bigl(\binom {i+j-2}{i-1}\Bigr)_{1\le i\le n,\ 1\le j\le n} = \begin{pmatrix} \binom {0}{0} & \binom {1}{0} & \cdots & \binom {n-1}{0} \\ \binom {1}{1} & \binom {2}{1} & \cdots & \binom {n}{1} \\ \vdots & \vdots & \ddots & \vdots \\ \binom {n-1}{n-1} & \binom {n}{n-1} & \cdots & \binom {2n-2}{n-1} \end{pmatrix}. \]

Then $\det A = 1$.

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theorem AlgebraicCombinatorics.CauchyBinet.det_pascal_matrix (n : ℕ) :

((pascalMatrix n).map fun (x : ℕ) => ↑x).det = 1

Factor hunting

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Theorem 5.75 Vandermonde determinant

Let $n \in \mathbb {N}$. Let $a_1, a_2, \ldots , a_n$ be $n$ elements of $K$. Then:

(a) We have

\[ \det \left( \left( a_i^{n-j} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = \prod _{1 \leq i {\lt} j \leq n} (a_i - a_j). \]

(b) We have

\[ \det \left( \left( a_j^{n-i} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = \prod _{1 \leq i {\lt} j \leq n} (a_i - a_j). \]

(c) We have

\[ \det \left( \left( a_i^{j-1} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = \prod _{1 \leq j {\lt} i \leq n} (a_i - a_j). \]

(d) We have

\[ \det \left( \left( a_j^{i-1} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = \prod _{1 \leq j {\lt} i \leq n} (a_i - a_j). \]

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theorem AlgebraicCombinatorics.Determinants.vandermonde_det_a {R : Type u_1} [CommRing R] {n : ℕ} (a : Fin n → R) :

(vandermondeMat a).det = ∏ i : Fin n, ∏ j ∈ Finset.Ioi i, (a i - a j)

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theorem AlgebraicCombinatorics.Determinants.vandermonde_det_b {R : Type u_1} [CommRing R] {n : ℕ} (a : Fin n → R) :

(vandermondeMat a).transpose.det = ∏ i : Fin n, ∏ j ∈ Finset.Ioi i, (a i - a j)

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theorem AlgebraicCombinatorics.Determinants.vandermonde_det_c {R : Type u_1} [CommRing R] {n : ℕ} (a : Fin n → R) :

(vandermondeMat' a).det = ∏ i : Fin n, ∏ j ∈ Finset.Iio i, (a i - a j)

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theorem AlgebraicCombinatorics.Determinants.vandermonde_det_d {R : Type u_1} [CommRing R] {n : ℕ} (a : Fin n → R) :

(vandermondeMat' a).transpose.det = ∏ i : Fin n, ∏ j ∈ Finset.Iio i, (a i - a j)

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Lemma 5.76

Let $n \in \mathbb {N}$. Consider the polynomial ring $\mathbb {Z}[x_1, x_2, \ldots , x_n]$ in $n$ indeterminates $x_1, x_2, \ldots , x_n$ with integer coefficients. In this ring, we have

\begin{equation} \det \left( \left( x_i^{n-j} \right)_{1 \leq i \leq n,\; 1 \leq j \leq n} \right) = \prod _{1 \leq i {\lt} j \leq n} (x_i - x_j). \label{eq.lem.det.vander.a.pol.1} \end{equation}

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theorem AlgebraicCombinatorics.Determinants.vandermonde_det_poly' (m : ℕ) :

(vandermondePolyMat m).det = vandermondePolyProd m

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Convention 5.78

Let $n \in \mathbb {N}$. Let $A$ be an $n \times n$-matrix. Let $i, j \in [n]$. Then, we set

\[ A_{\sim i, \sim j} := \operatorname {sub}_{[n] \setminus \{ i\} }^{[n] \setminus \{ j\} } A. \]

This is the $(n-1) \times (n-1)$-matrix obtained from $A$ by removing its $i$-th row and its $j$-th column.

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theorem AlgebraicCombinatorics.Determinants.det_sum_pow {R : Type u_1} [CommRing R] {n : ℕ} (x y : Fin n → R) :

(sumPowMat x y).det = (↑(∏ k : Fin n, (n - 1).choose ↑k) * ∏ i : Fin n, ∏ j ∈ Finset.Ioi i, (x i - x j)) * ∏ i : Fin n, ∏ j ∈ Finset.Ioi i, (y j - y i)

Laplace expansion

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Theorem 5.79 Laplace expansion

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix.

(a) For every $p \in [n]$, we have

\[ \det A = \sum _{q=1}^{n} (-1)^{p+q} A_{p,q} \det (A_{\sim p, \sim q}). \]

(b) For every $q \in [n]$, we have

\[ \det A = \sum _{p=1}^{n} (-1)^{p+q} A_{p,q} \det (A_{\sim p, \sim q}). \]

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theorem AlgebraicCombinatorics.Determinants.det_laplace_row {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 1)) (Fin (m + 1)) R) (p : Fin (m + 1)) :

A.det = ∑ q : Fin (m + 1), (-1) ^ (↑p + ↑q) * A p q * (submatrixRemove A p q).det

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theorem AlgebraicCombinatorics.Determinants.det_laplace_col {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 1)) (Fin (m + 1)) R) (q : Fin (m + 1)) :

A.det = ∑ p : Fin (m + 1), (-1) ^ (↑p + ↑q) * A p q * (submatrixRemove A p q).det

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Proposition 5.80

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix. Let $r \in [n]$.

(a) For every $p \in [n]$ satisfying $p \neq r$, we have

\[ 0 = \sum _{q=1}^{n} (-1)^{p+q} A_{r,q} \det (A_{\sim p, \sim q}). \]

(b) For every $q \in [n]$ satisfying $q \neq r$, we have

\[ 0 = \sum _{p=1}^{n} (-1)^{p+q} A_{p,r} \det (A_{\sim p, \sim q}). \]

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theorem AlgebraicCombinatorics.Determinants.det_laplace_row_zero {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 1)) (Fin (m + 1)) R) (p r : Fin (m + 1)) (hpr : p ≠ r) :

∑ q : Fin (m + 1), (-1) ^ (↑p + ↑q) * A r q * (submatrixRemove A p q).det = 0

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theorem AlgebraicCombinatorics.Determinants.det_laplace_col_zero {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 1)) (Fin (m + 1)) R) (q r : Fin (m + 1)) (hqr : q ≠ r) :

∑ p : Fin (m + 1), (-1) ^ (↑p + ↑q) * A p r * (submatrixRemove A p q).det = 0

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Definition 5.81 Adjugate matrix

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix. We define a new $n \times n$-matrix $\operatorname {adj} A \in K^{n \times n}$ by

\[ \operatorname {adj} A = \left( (-1)^{i+j} \det (A_{\sim j, \sim i}) \right)_{1 \leq i \leq n,\; 1 \leq j \leq n}. \]

This matrix $\operatorname {adj} A$ is called the adjugate (or classical adjoint) of the matrix $A$. Note the index swap: the $(i,j)$ entry involves removing row $j$ and column $i$.

The formalization shows that this definition equals Mathlib’s Matrix.adjugate, which is defined via $(\operatorname {adj} A)_{i,j} = \det (A')$ where $A'$ is the matrix $A$ with row $j$ replaced by the $i$-th standard basis vector.

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def AlgebraicCombinatorics.Determinants.adjugateMat {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 1)) (Fin (m + 1)) R) :

Matrix (Fin (m + 1)) (Fin (m + 1)) R

AlgebraicCombinatorics.Determinants.adjugateMat A = Matrix.of fun (i j : Fin (m + 1)) => (-1) ^ (↑i + ↑j) * (AlgebraicCombinatorics.Determinants.submatrixRemove A j i).det

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theorem AlgebraicCombinatorics.Determinants.adjugateMat_eq_adjugate {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 1)) (Fin (m + 1)) R) :

adjugateMat A = A.adjugate

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Convention 5.83

For any subset $I$ of $[n]$, we write $\widetilde{I} = [n] \setminus I$ for its complement, and $\operatorname {sum} S = \sum _{s \in S} s$ for any finite set $S$ of integers.

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theorem AlgebraicCombinatorics.Determinants.mul_adjugate' {R : Type u_1} [CommRing R] {n : ℕ} (A : Matrix (Fin n) (Fin n) R) :

A * A.adjugate = A.det • 1

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theorem AlgebraicCombinatorics.Determinants.adjugate_mul' {R : Type u_1} [CommRing R] {n : ℕ} (A : Matrix (Fin n) (Fin n) R) :

A.adjugate * A = A.det • 1

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Theorem 5.84 Laplace expansion along multiple rows/columns

Let $n \in \mathbb {N}$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix.

(a) For every subset $P$ of $[n]$, we have

\[ \det A = \sum _{\substack {Q \subseteq [n] \\ |Q| = |P|}} (-1)^{\operatorname {sum} P + \operatorname {sum} Q} \det \left( \operatorname {sub}_P^Q A \right) \det \left( \operatorname {sub}_{\widetilde{P}}^{\widetilde{Q}} A \right). \]

(b) For every subset $Q$ of $[n]$, we have

\[ \det A = \sum _{\substack {P \subseteq [n] \\ |P| = |Q|}} (-1)^{\operatorname {sum} P + \operatorname {sum} Q} \det \left( \operatorname {sub}_P^Q A \right) \det \left( \operatorname {sub}_{\widetilde{P}}^{\widetilde{Q}} A \right). \]

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theorem AlgebraicCombinatorics.Determinants.det_laplace_multi_row {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin m) (Fin m) R) (P : Finset (Fin m)) :

A.det = ∑ Q ∈ sameCardSubsets m P, (-1) ^ (P.sum Fin.val + Q.sum Fin.val) * submatrixDet A P Q * submatrixDet A Pᶜ Qᶜ

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theorem AlgebraicCombinatorics.Determinants.det_laplace_multi_col {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin m) (Fin m) R) (Q : Finset (Fin m)) :

A.det = ∑ P ∈ sameCardSubsets m Q, (-1) ^ (P.sum Fin.val + Q.sum Fin.val) * submatrixDet A P Q * submatrixDet A Pᶜ Qᶜ

Desnanot–Jacobi and Dodgson condensation

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Theorem 5.85 Desnanot–Jacobi formula, take 1

Let $n \in \mathbb {N}$ be such that $n \geq 2$. Let $A \in K^{n \times n}$ be an $n \times n$-matrix.

Let $A'$ be the $(n-2) \times (n-2)$-matrix

\[ \operatorname {sub}_{\{ 2, 3, \ldots , n-1\} }^{\{ 2, 3, \ldots , n-1\} } A = \left( A_{i+1, j+1} \right)_{1 \leq i \leq n-2,\; 1 \leq j \leq n-2}. \]

(This is what remains of $A$ when we remove the first row, the last row, the first column and the last column.) Then,

\begin{align*} \det A \cdot \det (A’) & = \det (A_{\sim 1, \sim 1}) \cdot \det (A_{\sim n, \sim n}) - \det (A_{\sim 1, \sim n}) \cdot \det (A_{\sim n, \sim 1}) \\ & = \det \begin{pmatrix} \det (A_{\sim 1, \sim 1}) & \det (A_{\sim 1, \sim n}) \\ \det (A_{\sim n, \sim 1}) & \det (A_{\sim n, \sim n}) \end{pmatrix}. \end{align*}

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theorem AlgebraicCombinatorics.Determinants.desnanot_jacobi {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 2)) (Fin (m + 2)) R) :

A.det * (innerSubmatrix A).det = (submatrixRemove A 0 0).det * (submatrixRemove A (Fin.last (m + 1)) (Fin.last (m + 1))).det - (submatrixRemove A 0 (Fin.last (m + 1))).det * (submatrixRemove A (Fin.last (m + 1)) 0).det

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theorem AlgebraicCombinatorics.Determinants.desnanot_jacobi_det2 {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 2)) (Fin (m + 2)) R) :

A.det * (innerSubmatrix A).det = !![(submatrixRemove A 0 0).det, (submatrixRemove A 0 (Fin.last (m + 1))).det; (submatrixRemove A (Fin.last (m + 1)) 0).det, (submatrixRemove A (Fin.last (m + 1)) (Fin.last (m + 1))).det].det

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Theorem 5.88

Let $n \in \mathbb {N}$ be such that $n \geq 2$. Let $p, q, u, v$ be four elements of $[n]$ such that $p {\lt} q$ and $u {\lt} v$. Let $A$ be an $n \times n$-matrix. Then,

\begin{align*} & \det A \cdot \det \left( \operatorname {sub}_{[n] \setminus \{ p, q\} }^{[n] \setminus \{ u, v\} } A \right) \\ & = \det (A_{\sim p, \sim u}) \cdot \det (A_{\sim q, \sim v}) - \det (A_{\sim p, \sim v}) \cdot \det (A_{\sim q, \sim u}). \end{align*}

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theorem AlgebraicCombinatorics.Determinants.desnanot_jacobi_general {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin (m + 2)) (Fin (m + 2)) R) (p q u v : Fin (m + 2)) (hpq : p < q) (huv : u < v) :

A.det * (submatrixRemove2 A p q u v hpq huv).det = (submatrixRemove A p u).det * (submatrixRemove A q v).det - (submatrixRemove A p v).det * (submatrixRemove A q u).det

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Theorem 5.89 Jacobi’s complementary minor theorem for adjugates

Let $n \in \mathbb {N}$. For any subset $I$ of $[n]$, let $\widetilde{I}$ denote the complement $[n] \setminus I$. Set $\operatorname {sum} S = \sum _{s \in S} s$ for any finite set $S$ of integers.

Let $A \in K^{n \times n}$ be an $n \times n$-matrix. Let $P$ and $Q$ be two subsets of $[n]$ such that $|P| = |Q| \geq 1$. Then,

\[ \det \left( \operatorname {sub}_P^Q (\operatorname {adj} A) \right) = (-1)^{\operatorname {sum} P + \operatorname {sum} Q} (\det A)^{|Q| - 1} \det \left( \operatorname {sub}_{\widetilde{Q}}^{\widetilde{P}} A \right). \]

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theorem AlgebraicCombinatorics.Determinants.jacobi_complementary_minor {R : Type u_1} [CommRing R] {m : ℕ} (A : Matrix (Fin m) (Fin m) R) (P Q : Finset (Fin m)) (hPQ : P.card = Q.card) (_hP : P.card ≥ 1) :

submatrixDet A.adjugate P Q = (-1) ^ (P.sum Fin.val + Q.sum Fin.val) * A.det ^ (Q.card - 1) * submatrixDet A Qᶜ Pᶜ

The Lindström–Gessel–Viennot lemma

Definitions

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Definition 5.120

We consider the infinite simple digraph with vertex set $\mathbb {Z}^{2}$ (so the vertices are pairs of integers) and arcs

\[ \left( i,j\right) \rightarrow \left( i+1,j\right) \ \ \ \ \ \ \ \ \ \ \text{for all }\left( i,j\right) \in \mathbb {Z}^{2} \]

and

\[ \left( i,j\right) \rightarrow \left( i,j+1\right) \ \ \ \ \ \ \ \ \ \ \text{for all }\left( i,j\right) \in \mathbb {Z}^{2}. \]

The arcs of the first form are called east-steps or right-steps; the arcs of the second form are called north-steps or up-steps.

The vertices of this digraph will be called lattice points or grid points or simply points.

The entire digraph will be denoted by $\mathbb {Z}^{2}$ and called the integer lattice or integer grid.

Any path is uniquely determined by its starting point and its step sequence.

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abbrev LGV1.LatticePoint :

LGV1.LatticePoint = (ℤ × ℤ)

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inductive LGV1.LatticeStep :

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def LGV1.IntegerLatticeAdj (p q : LatticePoint) :

LGV1.IntegerLatticeAdj p q = (q = (p.1 + 1, p.2) ∨ q = (p.1, p.2 + 1))

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def LGV1.integerLattice :

Digraph LatticePoint

LGV1.integerLattice = { Adj := LGV1.IntegerLatticeAdj }

Counting paths from $\left( a,b\right) $ to $\left( c,d\right) $

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Proposition 5.129

Let $\left( a,b\right) \in \mathbb {Z}^{2}$ and $\left( c,d\right) \in \mathbb {Z}^{2}$ be two points. Then,

\[ \left( \text{\# of paths from }\left( a,b\right) \text{ to }\left( c,d\right) \right) =\begin{cases} \binom {c+d-a-b}{c-a}, & \text{if }c+d\geq a+b;\\ 0, & \text{if }c+d{\lt}a+b. \end{cases} \]

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theorem LGV1.numPaths_eq_card (a b : LatticePoint) (hfin : (pathsFromTo a b).Finite) :

numPaths a b = hfin.toFinset.card

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def LGV1.numPaths (a b : LatticePoint) :

LGV1.numPaths a b = if a.1 ≤ b.1 ∧ a.2 ≤ b.2 then (b.1 - a.1 + (b.2 - a.2)).toNat.choose (b.1 - a.1).toNat else 0

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theorem LGV1.no_paths_when_sum_decreases (a b : LatticePoint) (h : b.coordSum < a.coordSum) :

pathsFromTo a b = ∅

Path tuples, nipats and ipats

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Definition 5.130

Let $k\in \mathbb {N}$.

(a) A $k$-vertex means a $k$-tuple of lattice points.

(b) If $\mathbf{A}=\left( A_{1},A_{2},\ldots ,A_{k}\right) $ is a $k$-vertex, and if $\sigma \in S_{k}$ is a permutation, then $\sigma \left( \mathbf{A}\right) $ shall denote the $k$-vertex $\left( A_{\sigma \left( 1\right) },A_{\sigma \left( 2\right) },\ldots ,A_{\sigma \left( k\right) }\right) $.

(c) If $\mathbf{A}=\left( A_{1},A_{2},\ldots ,A_{k}\right) $ and $\mathbf{B}=\left( B_{1},B_{2},\ldots ,B_{k}\right) $ are two $k$-vertices, then a path tuple from $\mathbf{A}$ to $\mathbf{B}$ means a $k$-tuple $\left( p_{1},p_{2},\ldots ,p_{k}\right) $, where each $p_{i}$ is a path from $A_{i}$ to $B_{i}$.

(d) A path tuple $\left( p_{1},p_{2},\ldots ,p_{k}\right) $ is said to be non-intersecting if no two of the paths $p_{1},p_{2},\ldots ,p_{k}$ have any vertex in common. We abbreviate “non-intersecting path tuple” as nipat.

(e) A path tuple $\left( p_{1},p_{2},\ldots ,p_{k}\right) $ is said to be intersecting if it is not non-intersecting (i.e., if two of its paths have a vertex in common). We abbreviate “intersecting path tuple” as ipat.

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abbrev LGV1.kVertex (k : ℕ) :

LGV1.kVertex k = (Fin k → LGV1.LatticePoint)

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def LGV1.kVertex.permute {k : ℕ} (v : kVertex k) (σ : Equiv.Perm (Fin k)) :

kVertex k

v.permute σ i = v (σ i)

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structure LGV1.PathTuple (k : ℕ) (A B : kVertex k) :

paths : Fin k → LatticePath
The i-th path in the tuple
valid (i : Fin k) : (self.paths i).isPathFromTo (A i) (B i)
Each path goes from A_i to B_i

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def LGV1.PathTuple.isNonIntersecting {k : ℕ} {A B : kVertex k} (pt : PathTuple k A B) :

pt.isNonIntersecting = ∀ (i j : Fin k), i ≠ j → Disjoint (pt.verticesOf i) (pt.verticesOf j)

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def LGV1.PathTuple.isIntersecting {k : ℕ} {A B : kVertex k} (pt : PathTuple k A B) :

pt.isIntersecting = ¬pt.isNonIntersecting

The LGV lemma for two paths

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Proposition 5.145 LGV lemma for two paths

Let $\left( A,A^{\prime }\right) $ and $\left( B,B^{\prime }\right) $ be two $2$-vertices (i.e., let $A,A^{\prime },B,B^{\prime }$ be four lattice points). Then,

\begin{align*} & \det \left( \begin{array}[c]{cc}\left( \text{\# of paths from }A\text{ to }B\right) & \left( \text{\# of paths from }A\text{ to }B^{\prime }\right) \\ \left( \text{\# of paths from }A^{\prime }\text{ to }B\right) & \left( \text{\# of paths from }A^{\prime }\text{ to }B^{\prime }\right) \end{array} \right) \\ & =\left( \text{\# of nipats from }\left( A,A^{\prime }\right) \text{ to }\left( B,B^{\prime }\right) \right) \\ & \ \ \ \ \ \ \ \ \ \ -\left( \text{\# of nipats from }\left( A,A^{\prime }\right) \text{ to }\left( B^{\prime },B\right) \right) . \end{align*}

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theorem LGV1.lgv_two_paths (A A' B B' : LatticePoint) :

(pathMatrix2 A A' B B').det = ↑(numNipats2 A A' B B') - ↑(numNipats2 A A' B' B)

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Proposition 5.146 baby Jordan curve theorem

Let $A$, $B$, $A^{\prime }$ and $B^{\prime }$ be four lattice points satisfying

\begin{align*} \operatorname {x}\left( A^{\prime }\right) & \leq \operatorname {x}\left( A\right) ,\ \ \ \ \ \ \ \ \ \ \operatorname {y}\left( A^{\prime }\right) \geq \operatorname {y}\left( A\right) ,\\ \operatorname {x}\left( B^{\prime }\right) & \leq \operatorname {x}\left( B\right) ,\ \ \ \ \ \ \ \ \ \ \operatorname {y}\left( B^{\prime }\right) \geq \operatorname {y}\left( B\right) . \end{align*}

(That is, $A'$ is weakly northwest of $A$, and $B'$ is weakly northwest of $B$.)

Let $p$ be any path from $A$ to $B^{\prime }$. Let $p^{\prime }$ be any path from $A^{\prime }$ to $B$. Then, $p$ and $p^{\prime }$ have a vertex in common.

In particular, there are no nipats from $(A, A')$ to $(B', B)$.

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theorem LGV1.baby_jordan_curve (A A' B B' : LatticePoint) (hA : isWeaklyNorthwestOf A' A) (hB : isWeaklyNorthwestOf B' B) (p p' : LatticePath) (hp : p.isPathFromTo A B') (hp' : p'.isPathFromTo A' B) :

∃ v ∈ p.vertices A, v ∈ p'.vertices A'

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theorem LGV1.no_nipats_under_nw (A A' B B' : LatticePoint) (hA : isWeaklyNorthwestOf A' A) (hB : isWeaklyNorthwestOf B' B) :

numNipats2 A A' B' B = 0

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def LGV1.isWeaklyNorthwestOf (p q : LatticePoint) :

LGV1.isWeaklyNorthwestOf p q = (p.x ≤ q.x ∧ p.y ≥ q.y)

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Corollary 5.147

Let $n,k\in \mathbb {N}$. Then, $\binom {n}{k}^{2}\geq \binom {n}{k-1}\cdot \binom {n}{k+1}$.

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theorem LGV1.binom_log_concave (n k : ℕ) (hk : 1 ≤ k) :

n.choose k * n.choose k ≥ n.choose (k - 1) * n.choose (k + 1)

The LGV lemma for $k$ paths

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Proposition 5.153 LGV lemma, lattice counting version

Let $k\in \mathbb {N}$. Let $\mathbf{A}=\left( A_{1},A_{2},\ldots ,A_{k}\right) $ and $\mathbf{B}=\left( B_{1},B_{2},\ldots ,B_{k}\right) $ be two $k$-vertices. Then,

\begin{align*} & \det \left( \left( \text{\# of paths from }A_{i}\text{ to }B_{j}\right) _{1\leq i\leq k,\ 1\leq j\leq k}\right) \\ & =\sum _{\sigma \in S_{k}}\left( -1\right) ^{\sigma }\left( \text{\# of nipats from }\mathbf{A}\text{ to }\sigma \left( \mathbf{B}\right) \right) . \end{align*}

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theorem LGV1.lgv_k_paths {k : ℕ} (A B : kVertex k) :

(pathMatrixK A B).det = ∑ σ : Equiv.Perm (Fin k), ↑(Equiv.Perm.sign σ) * ↑(numNipatsK A B σ)

The weighted version

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Theorem 5.154 LGV lemma, lattice weight version

Let $K$ be a commutative ring.

For each arc $a$ of the digraph $\mathbb {Z}^{2}$, let $w(a)$ be an element of $K$. We call $w(a)$ the weight of $a$.

For each path $p$ of $\mathbb {Z}^{2}$, define the weight $w(p)$ of $p$ by

\[ w(p) := \prod _{a \text{ is an arc of } p} w(a). \]

For each path tuple $\mathbf{p} = (p_1, p_2, \ldots , p_k)$, define the weight $w(\mathbf{p})$ of $\mathbf{p}$ by

\[ w(\mathbf{p}) := w(p_1)\, w(p_2) \cdots w(p_k). \]

Let $k \in \mathbb {N}$. Let $\mathbf{A} = (A_1, A_2, \ldots , A_k)$ and $\mathbf{B} = (B_1, B_2, \ldots , B_k)$ be two $k$-vertices. Then,

\[ \det \! \left(\left(\sum _{p : A_i \to B_j} w(p)\right)_{1 \le i \le k,\; 1 \le j \le k}\right) = \sum _{\sigma \in S_k} (-1)^{\sigma } \sum _{\substack {\mathbf{p} \text{ is a nipat} \\ \text{from } \mathbf{A} \text{ to } \sigma (\mathbf{B})}} w(\mathbf{p}). \]

Here, “$p : A_i \to B_j$” means “$p$ is a path from $A_i$ to $B_j$”.

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theorem LGV.lgv_weighted_lattice {K : Type u_2} [CommRing K] {k : ℕ} (w : ArcWeight integerLattice K) (A B : kVertex (ℤ × ℤ) k) :

(pathWeightMatrix integerLattice_pathFinite w A B).det = ∑ σ : Equiv.Perm (Fin k), Equiv.Perm.sign σ • nipatWeightSum integerLattice_pathFinite w A (permuteKVertex σ B) σ

Generalization to acyclic digraphs

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Theorem 5.155 LGV lemma, digraph weight version

Let $K$ be a commutative ring.

Let $D$ be a path-finite (but possibly infinite) acyclic digraph.

For each arc $a$ of $D$, let $w(a) \in K$. For each path $p$ of $D$, define $w(p) := \prod _{a \text{ is an arc of } p} w(a)$. For each path tuple $\mathbf{p} = (p_1, \ldots , p_k)$, define $w(\mathbf{p}) := w(p_1) \cdots w(p_k)$.

Let $k \in \mathbb {N}$. Let $\mathbf{A} = (A_1, \ldots , A_k)$ and $\mathbf{B} = (B_1, \ldots , B_k)$ be two $k$-tuples of vertices of $D$. Then,

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theorem LGV.lgv_weighted_digraph {K : Type u_2} [CommRing K] {V : Type u_4} [DecidableEq V] {D : SimpleDigraph V} (hpf : D.IsPathFinite) (hac : D.IsAcyclic) {k : ℕ} (w : ArcWeight D K) (A B : kVertex V k) :

(pathWeightMatrix hpf w A B).det = ∑ σ : Equiv.Perm (Fin k), Equiv.Perm.sign σ • nipatWeightSum hpf w A (permuteKVertex σ B) σ

The nonpermutable case

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Corollary 5.175 LGV lemma, nonpermutable lattice weight version

Consider the setting of Theorem 5.154, but additionally assume that

\begin{align} \operatorname {x}(A_1) & \ge \operatorname {x}(A_2) \ge \cdots \ge \operatorname {x}(A_k); \label{eq.cor.lgv.kpaths.wt-np.xA} \\ \operatorname {y}(A_1) & \le \operatorname {y}(A_2) \le \cdots \le \operatorname {y}(A_k); \label{eq.cor.lgv.kpaths.wt-np.yA} \\ \operatorname {x}(B_1) & \ge \operatorname {x}(B_2) \ge \cdots \ge \operatorname {x}(B_k); \label{eq.cor.lgv.kpaths.wt-np.xB} \\ \operatorname {y}(B_1) & \le \operatorname {y}(B_2) \le \cdots \le \operatorname {y}(B_k). \label{eq.cor.lgv.kpaths.wt-np.yB} \end{align}

Here, $\operatorname {x}(P)$ and $\operatorname {y}(P)$ denote the two coordinates of any point $P \in \mathbb {Z}^2$.

Then, there are no nipats from $\mathbf{A}$ to $\sigma (\mathbf{B})$ when $\sigma \in S_k$ is not the identity permutation $\operatorname {id} \in S_k$. Therefore, the claim of Theorem 5.154 simplifies to

\begin{equation} \det \! \left(\left(\sum _{p : A_i \to B_j} w(p)\right)_{1 \le i \le k,\; 1 \le j \le k}\right) = \sum _{\substack {\mathbf{p} \text{ is a nipat} \\ \text{from } \mathbf{A} \text{ to } \mathbf{B}}} w(\mathbf{p}). \label{eq.cor.lgv.kpaths.wt-np.claim} \end{equation}

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theorem LGV.lgv_nonpermutable {K : Type u_2} [CommRing K] {k : ℕ} (w : ArcWeight integerLattice K) (A B : kVertex (ℤ × ℤ) k) (hxA : xDecreasing A) (hyA : yIncreasing A) (hxB : xDecreasing B) (hyB : yIncreasing B) :

(pathWeightMatrix integerLattice_pathFinite w A B).det = nipatWeightSum integerLattice_pathFinite w A B (Equiv.refl (Fin k))

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Corollary 5.181

Let $k \in \mathbb {N}$. Let $a_1, a_2, \ldots , a_k$ and $b_1, b_2, \ldots , b_k$ be nonnegative integers such that

\[ a_1 \ge a_2 \ge \cdots \ge a_k \qquad \text{and} \qquad b_1 \ge b_2 \ge \cdots \ge b_k. \]

Then,

\[ \det \! \left(\left(\binom {a_i}{b_j}\right)_{1 \le i \le k,\; 1 \le j \le k}\right) \ge 0. \]

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theorem LGV.binom_det_nonneg {k : ℕ} (a b : Fin k → ℕ) (ha : ∀ (i j : Fin k), i ≤ j → a j ≤ a i) (hb : ∀ (i j : Fin k), i ≤ j → b j ≤ b i) :

0 ≤ (Matrix.of fun (i j : Fin k) => ↑((a i).choose (b j))).det

Definitions and examples of symmetric polynomials

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Definition 6.2

(a) Let $\mathcal{P}$ be the polynomial ring $K[x_1, x_2, \ldots , x_N]$ in $N$ variables over $K$. This is not just a ring; it is a commutative $K$-algebra.

(b) The symmetric group $S_N$ acts on the set $\mathcal{P}$ according to the formula

\[ \sigma \cdot f = f[x_{\sigma (1)}, x_{\sigma (2)}, \ldots , x_{\sigma (N)}] \quad \text{for any } \sigma \in S_N \text{ and any } f \in \mathcal{P}. \]

Here, $f[a_1, a_2, \ldots , a_N]$ means the result of substituting $a_1, a_2, \ldots , a_N$ for the indeterminates $x_1, x_2, \ldots , x_N$ in a polynomial $f \in \mathcal{P}$.

Roughly speaking, the group $S_N$ is thus acting on $\mathcal{P}$ by permuting variables: A permutation $\sigma \in S_N$ transforms a polynomial $f$ by substituting $x_{\sigma (i)}$ for each $x_i$.

Note that this action of $S_N$ on $\mathcal{P}$ is a well-defined group action (as we will see in Proposition 6.3 below).

(c) A polynomial $f \in \mathcal{P}$ is said to be symmetric if it satisfies

\[ \sigma \cdot f = f \quad \text{for all } \sigma \in S_N. \]

(d) We let $\mathcal{S}$ be the set of all symmetric polynomials $f \in \mathcal{P}$.

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abbrev AlgebraicCombinatorics.SymmetricPolynomials.P (K : Type u_2) [CommRing K] (N : ℕ) :

Type u_2

AlgebraicCombinatorics.SymmetricPolynomials.P K N = MvPolynomial (Fin N) K

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noncomputable def AlgebraicCombinatorics.SymmetricPolynomials.permAction {K : Type u_1} [CommRing K] {N : ℕ} (σ : Equiv.Perm (Fin N)) (f : P K N) :

P K N

AlgebraicCombinatorics.SymmetricPolynomials.permAction σ f = (MvPolynomial.rename ⇑σ) f

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def AlgebraicCombinatorics.SymmetricPolynomials.IsSymm {K : Type u_1} [CommRing K] {N : ℕ} (f : P K N) :

AlgebraicCombinatorics.SymmetricPolynomials.IsSymm f = MvPolynomial.IsSymmetric f

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abbrev AlgebraicCombinatorics.SymmetricPolynomials.S (K : Type u_2) [CommRing K] (N : ℕ) :

Subalgebra K (P K N)

AlgebraicCombinatorics.SymmetricPolynomials.S K N = MvPolynomial.symmetricSubalgebra (Fin N) K

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Proposition 6.3

The action of $S_N$ on $\mathcal{P}$ is a well-defined group action. In other words, the following holds:

(a) We have $\operatorname {id}_{[N]} \cdot f = f$ for every $f \in \mathcal{P}$.

(b) We have $(\sigma \tau ) \cdot f = \sigma \cdot (\tau \cdot f)$ for every $\sigma , \tau \in S_N$ and $f \in \mathcal{P}$.

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theorem AlgebraicCombinatorics.SymmetricPolynomials.permAction_id {K : Type u_1} [CommRing K] {N : ℕ} (f : P K N) :

permAction 1 f = f

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theorem AlgebraicCombinatorics.SymmetricPolynomials.permAction_mul {K : Type u_1} [CommRing K] {N : ℕ} (σ τ : Equiv.Perm (Fin N)) (f : P K N) :

permAction (σ * τ) f = permAction σ (permAction τ f)

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noncomputable instance AlgebraicCombinatorics.SymmetricPolynomials.permMulAction {K : Type u_1} [CommRing K] {N : ℕ} :

MulAction (Equiv.Perm (Fin N)) (P K N)

AlgebraicCombinatorics.SymmetricPolynomials.permMulAction = { smul := AlgebraicCombinatorics.SymmetricPolynomials.permAction, mul_smul := ⋯, one_smul := ⋯ }

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Proposition 6.4

The group $S_N$ acts on $\mathcal{P}$ by $K$-algebra automorphisms. In other words, for each $\sigma \in S_N$, the map

\begin{align*} \mathcal{P} & \to \mathcal{P}, \\ f & \mapsto \sigma \cdot f \end{align*}

is a $K$-algebra automorphism of $\mathcal{P}$.

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noncomputable def AlgebraicCombinatorics.SymmetricPolynomials.permAutomorphism {K : Type u_1} [CommRing K] {N : ℕ} (σ : Equiv.Perm (Fin N)) :

P K N ≃ₐ[K] P K N

AlgebraicCombinatorics.SymmetricPolynomials.permAutomorphism σ = MvPolynomial.renameEquiv K σ

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noncomputable def AlgebraicCombinatorics.SymmetricPolynomials.permAutomorphismHom {K : Type u_1} [CommRing K] {N : ℕ} :

Equiv.Perm (Fin N) →* P K N ≃ₐ[K] P K N

One or more equations did not get rendered due to their size.

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Theorem 6.5

The subset $\mathcal{S}$ is a $K$-subalgebra of $\mathcal{P}$.

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def AlgebraicCombinatorics.SymmetricPolynomials.S_subalgebra {K : Type u_1} [CommRing K] {N : ℕ} :

Subalgebra K (P K N)

AlgebraicCombinatorics.SymmetricPolynomials.S_subalgebra = AlgebraicCombinatorics.SymmetricPolynomials.S K N

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Definition 6.6

The $K$-subalgebra $\mathcal{S}$ of $\mathcal{P}$ is called the ring of symmetric polynomials in $N$ variables over $K$.

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abbrev AlgebraicCombinatorics.SymmetricPolynomials.S (K : Type u_2) [CommRing K] (N : ℕ) :

Subalgebra K (P K N)

AlgebraicCombinatorics.SymmetricPolynomials.S K N = MvPolynomial.symmetricSubalgebra (Fin N) K

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Definition 6.7

(a) A monomial is an expression of the form $x_1^{a_1} x_2^{a_2} \cdots x_N^{a_N}$ with $a_1, a_2, \ldots , a_N \in \mathbb {N}$.

(b) The degree $\deg \mathfrak {m}$ of a monomial $\mathfrak {m} = x_1^{a_1} x_2^{a_2} \cdots x_N^{a_N}$ is defined to be $a_1 + a_2 + \cdots + a_N \in \mathbb {N}$.

(c) A monomial $\mathfrak {m} = x_1^{a_1} x_2^{a_2} \cdots x_N^{a_N}$ is said to be squarefree if $a_1, a_2, \ldots , a_N \in \{ 0,1\} $. (This is saying that no square or higher power of an indeterminate appears in $\mathfrak {m}$; thus the name “squarefree”.)

(d) A monomial $\mathfrak {m} = x_1^{a_1} x_2^{a_2} \cdots x_N^{a_N}$ is said to be primal if there is at most one $i \in [N]$ satisfying $a_i {\gt} 0$. (This is saying that the monomial $\mathfrak {m}$ contains no two distinct indeterminates. Thus, a primal monomial is just $1$ or a power of an indeterminate.)

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abbrev AlgebraicCombinatorics.SymmetricPolynomials.Monomial (N : ℕ) :

AlgebraicCombinatorics.SymmetricPolynomials.Monomial N = (Fin N →₀ ℕ)

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def AlgebraicCombinatorics.SymmetricPolynomials.Monomial.degree {N : ℕ} (m : Monomial N) :

m.degree = Finsupp.sum m fun (x : Fin N) (a : ℕ) => a

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def AlgebraicCombinatorics.SymmetricPolynomials.Monomial.IsSquarefree {N : ℕ} (m : Monomial N) :

m.IsSquarefree = ∀ (i : Fin N), m i ≤ 1

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def AlgebraicCombinatorics.SymmetricPolynomials.Monomial.IsPrimal {N : ℕ} (m : Monomial N) :

m.IsPrimal = (m.support.card ≤ 1)

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Definition 6.8

(a) For each $n \in \mathbb {Z}$, define a symmetric polynomial $e_n \in \mathcal{S}$ by

\[ e_n = \sum _{\substack {(i_1, i_2, \ldots , i_n) \in [N]^n; \\ i_1 {\lt} i_2 {\lt} \cdots {\lt} i_n}} x_{i_1} x_{i_2} \cdots x_{i_n} = (\text{sum of all squarefree monomials of degree } n). \]

This $e_n$ is called the $n$-th elementary symmetric polynomial in $x_1, x_2, \ldots , x_N$.

(b) For each $n \in \mathbb {Z}$, define a symmetric polynomial $h_n \in \mathcal{S}$ by

\[ h_n = \sum _{\substack {(i_1, i_2, \ldots , i_n) \in [N]^n; \\ i_1 \leq i_2 \leq \cdots \leq i_n}} x_{i_1} x_{i_2} \cdots x_{i_n} = (\text{sum of all monomials of degree } n). \]

This $h_n$ is called the $n$-th complete homogeneous symmetric polynomial in $x_1, x_2, \ldots , x_N$.

(c) For each $n \in \mathbb {Z}$, define a symmetric polynomial $p_n \in \mathcal{S}$ by

\begin{align*} p_n & = \begin{cases} x_1^n + x_2^n + \cdots + x_N^n, & \text{if } n {\gt} 0; \\ 1, & \text{if } n = 0; \\ 0, & \text{if } n {\lt} 0 \end{cases}\\ & = (\text{sum of all primal monomials of degree } n). \end{align*}

This $p_n$ is called the $n$-th power sum in $x_1, x_2, \ldots , x_N$.

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noncomputable abbrev AlgebraicCombinatorics.SymmetricPolynomials.e {K : Type u_1} [CommRing K] {N : ℕ} (n : ℕ) :

P K N

AlgebraicCombinatorics.SymmetricPolynomials.e n = MvPolynomial.esymm (Fin N) K n

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noncomputable abbrev AlgebraicCombinatorics.SymmetricPolynomials.h {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] (n : ℕ) :

P K N

AlgebraicCombinatorics.SymmetricPolynomials.h n = MvPolynomial.hsymm (Fin N) K n

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noncomputable abbrev AlgebraicCombinatorics.SymmetricPolynomials.p {K : Type u_1} [CommRing K] {N : ℕ} (n : ℕ) :

P K N

AlgebraicCombinatorics.SymmetricPolynomials.p n = MvPolynomial.psum (Fin N) K n

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Proposition 6.9

For each integer $n {\gt} N$, we have $e_n = 0$.

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theorem AlgebraicCombinatorics.SymmetricPolynomials.e_eq_zero_of_gt {K : Type u_1} [CommRing K] {N n : ℕ} (hn : N < n) :

e n = 0

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Theorem 6.12 Newton–Girard formulas

For any positive integer $n$, we have

\begin{align} \sum _{j=0}^{n} (-1)^j e_j h_{n-j} & = 0; \label{eq.thm.sf.NG.eh} \\ \sum _{j=1}^{n} (-1)^{j-1} e_{n-j} p_j & = n e_n; \label{eq.thm.sf.NG.ep} \\ \sum _{j=1}^{n} h_{n-j} p_j & = n h_n. \label{eq.thm.sf.NG.hp} \end{align}

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theorem AlgebraicCombinatorics.SymmetricPolynomials.newtonGirard_eh {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] (n : ℕ) (hn : 0 < n) :

∑ j ∈ Finset.range (n + 1), (-1) ^ j * e j * h (n - j) = 0

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theorem AlgebraicCombinatorics.SymmetricPolynomials.newtonGirard_hp {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] (n : ℕ) (_hn : 0 < n) :

∑ j ∈ Finset.range n, h (n - 1 - j) * p (j + 1) = ↑n * h n

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theorem AlgebraicCombinatorics.SymmetricPolynomials.newtonGirard_esymm {K : Type u_1} [CommRing K] {N : ℕ} (k : ℕ) :

↑k * e k = (-1) ^ (k + 1) * ∑ a ∈ Finset.antidiagonal k with a.1 < k, (-1) ^ a.1 * e a.1 * p a.2

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Proposition 6.10

(a) In the polynomial ring $\mathcal{P}[t]$, we have

\[ \prod _{i=1}^{N} (1 - t x_i) = \sum _{n \in \mathbb {N}} (-1)^n t^n e_n. \]

(b) In the polynomial ring $\mathcal{P}[u,v]$, we have

\[ \prod _{i=1}^{N} (u - v x_i) = \sum _{n=0}^{N} (-1)^n u^{N-n} v^n e_n. \]

(c) In the FPS ring $\mathcal{P}[[t]]$, we have

\[ \prod _{i=1}^{N} \frac{1}{1 - t x_i} = \sum _{n \in \mathbb {N}} t^n h_n. \]

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theorem AlgebraicCombinatorics.SymmetricPolynomials.esymm_genfunc {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] :

∏ i : Fin N, (1 - Polynomial.X * Polynomial.C (MvPolynomial.X i)) = ∑ n ∈ Finset.range (N + 1), (-1) ^ n * Polynomial.X ^ n * Polynomial.C (e n)

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theorem AlgebraicCombinatorics.SymmetricPolynomials.esymm_genfunc_two_var {K : Type u_1} [CommRing K] {N : ℕ} :

∏ i : Fin N, (MvPolynomial.X 0 - MvPolynomial.X 1 * MvPolynomial.C (MvPolynomial.X i)) = ∑ n ∈ Finset.range (N + 1), (-1) ^ n * MvPolynomial.X 0 ^ (N - n) * MvPolynomial.X 1 ^ n * MvPolynomial.C (e n)

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theorem AlgebraicCombinatorics.SymmetricPolynomials.hsymm_genfunc {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] :

have E := ∏ i : Fin N, (1 - Polynomial.X * Polynomial.C (MvPolynomial.X i)); (PowerSeries.mk fun (n : ℕ) => h n) * ↑E = 1

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Theorem 6.13 Fundamental Theorem of Symmetric Polynomials

(a) The elementary symmetric polynomials $e_1, e_2, \ldots , e_N$ are algebraically independent (over $K$) and generate the $K$-algebra $\mathcal{S}$.

In other words, each $f \in \mathcal{S}$ can be uniquely written as a polynomial in $e_1, e_2, \ldots , e_N$.

In yet other words, the map

\begin{align*} K[y_1, y_2, \ldots , y_N] & \to \mathcal{S}, \\ g & \mapsto g[e_1, e_2, \ldots , e_N] \end{align*}

is a $K$-algebra isomorphism.

(b) The complete homogeneous symmetric polynomials $h_1, h_2, \ldots , h_N$ are algebraically independent (over $K$) and generate the $K$-algebra $\mathcal{S}$.

In other words, each $f \in \mathcal{S}$ can be uniquely written as a polynomial in $h_1, h_2, \ldots , h_N$.

In yet other words, the map

\begin{align*} K[y_1, y_2, \ldots , y_N] & \to \mathcal{S}, \\ g & \mapsto g[h_1, h_2, \ldots , h_N] \end{align*}

is a $K$-algebra isomorphism.

(c) Now assume that $K$ is a commutative $\mathbb {Q}$-algebra (e.g., a field of characteristic $0$). Then, the power sums $p_1, p_2, \ldots , p_N$ are algebraically independent (over $K$) and generate the $K$-algebra $\mathcal{S}$.

In other words, each $f \in \mathcal{S}$ can be uniquely written as a polynomial in $p_1, p_2, \ldots , p_N$.

In yet other words, the map

\begin{align*} K[y_1, y_2, \ldots , y_N] & \to \mathcal{S}, \\ g & \mapsto g[p_1, p_2, \ldots , p_N] \end{align*}

is a $K$-algebra isomorphism.

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noncomputable def AlgebraicCombinatorics.SymmetricPolynomials.esymmAlgEquiv' {K : Type u_1} [CommRing K] {N : ℕ} :

MvPolynomial (Fin N) K ≃ₐ[K] ↥(S K N)

AlgebraicCombinatorics.SymmetricPolynomials.esymmAlgEquiv' = MvPolynomial.esymmAlgEquiv (Fin N) K ⋯

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theorem AlgebraicCombinatorics.SymmetricPolynomials.esymm_algebraicIndependent {K : Type u_1} [CommRing K] {N : ℕ} :

AlgebraicIndependent K fun (i : Fin N) => MvPolynomial.esymm (Fin N) K (↑i + 1)

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theorem AlgebraicCombinatorics.SymmetricPolynomials.esymm_generates_symmetric {K : Type u_1} [CommRing K] {N : ℕ} (f : ↥(S K N)) :

∃! g : MvPolynomial (Fin N) K, (MvPolynomial.esymmAlgHom (Fin N) K N) g = f

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theorem AlgebraicCombinatorics.SymmetricPolynomials.hsymm_algebraicIndependent {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] [IsDomain K] :

AlgebraicIndependent K fun (i : Fin N) => MvPolynomial.hsymm (Fin N) K (↑i + 1)

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theorem AlgebraicCombinatorics.SymmetricPolynomials.hsymm_generates_symmetric {K : Type u_1} [CommRing K] {N : ℕ} [DecidableEq (Fin N)] [IsDomain K] (f : ↥(S K N)) :

∃! g : MvPolynomial (Fin N) K, (MvPolynomial.aeval fun (i : Fin N) => MvPolynomial.hsymm (Fin N) K (↑i + 1)) g = ↑f

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theorem AlgebraicCombinatorics.SymmetricPolynomials.psum_algebraicIndependent {K : Type u_1} [CommRing K] {N : ℕ} [Algebra ℚ K] [IsDomain K] :

AlgebraicIndependent K fun (i : Fin N) => MvPolynomial.psum (Fin N) K (↑i + 1)

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theorem AlgebraicCombinatorics.SymmetricPolynomials.psum_generates_symmetric {K : Type u_1} [CommRing K] {N : ℕ} [Algebra ℚ K] [IsDomain K] (f : ↥(S K N)) :

∃! g : MvPolynomial (Fin N) K, (MvPolynomial.aeval fun (i : Fin N) => MvPolynomial.psum (Fin N) K (↑i + 1)) g = ↑f

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Lemma 6.14

For each $i \in [N-1]$, let $s_i \in S_N$ denote the simple transposition that swaps $i$ and $i+1$.

Let $f \in \mathcal{P}$. Assume that

\begin{equation} s_k \cdot f = f \quad \text{for each } k \in [N-1]. \label{eq.lem.sf.simples-enough.ass} \end{equation}

Then, the polynomial $f$ is symmetric.

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theorem AlgebraicCombinatorics.SymmetricPolynomials.isSymm_iff_simpleTranspositions {K : Type u_1} [CommRing K] {N : ℕ} (f : P K N) :

IsSymm f ↔ ∀ (k : Fin (N - 1)), permAction (simpleTransposition k) f = f

$N$-partitions and monomial symmetric polynomials

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Definition 6.30

An $N$-partition will mean a weakly decreasing $N$-tuple of nonnegative integers. In other words, an $N$-partition means an $N$-tuple $\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{N}\right) \in \mathbb {N}^{N}$ with $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{N}$.

The size (or weight) of an $N$-partition $\lambda = (\lambda _1, \lambda _2, \ldots , \lambda _N)$ is defined to be $|\lambda | := \lambda _1 + \lambda _2 + \cdots + \lambda _N$.

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abbrev AlgebraicCombinatorics.SymmetricFunctions.NPartition (N : ℕ) :

AlgebraicCombinatorics.SymmetricFunctions.NPartition N = NPartition N

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Proposition 6.31

There is a bijection

\begin{align*} \left\{ \text{partitions of length }\leq N\right\} & \rightarrow \left\{ N\text{-partitions}\right\} ,\\ \left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{\ell }\right) & \mapsto \left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{\ell },\underbrace{0,0,\ldots ,0}_{N-\ell \text{ zeroes}}\right) . \end{align*}

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noncomputable def AlgebraicCombinatorics.SymmetricFunctions.NPartition.equivPartition {N : ℕ} (n : ℕ) :

{ p : n.Partition // p.parts.card ≤ N } ≃ { mu : NPartition N // NPartition.size mu = n }

One or more equations did not get rendered due to their size.

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Definition 6.44

Let $a=\left( a_{1},a_{2},\ldots ,a_{N}\right) \in \mathbb {N}^{N}$. Then:

(a) We let $x^{a}$ denote the monomial $x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{N}^{a_{N}}$.

(b) We let $\operatorname *{sort}a$ mean the $N$-partition obtained from $a$ by sorting the entries of $a$ in weakly decreasing order.

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noncomputable def AlgebraicCombinatorics.SymmetricFunctions.monomialExp {R : Type u_1} [CommSemiring R] {N : ℕ} (a : Fin N → ℕ) :

MvPolynomial (Fin N) R

AlgebraicCombinatorics.SymmetricFunctions.monomialExp a = ∏ i : Fin N, MvPolynomial.X i ^ a i

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def AlgebraicCombinatorics.SymmetricFunctions.sortTuple {N : ℕ} (a : Fin N → ℕ) :

NPartition N

One or more equations did not get rendered due to their size.

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Definition 6.58

Let $\lambda $ be any $N$-partition. Then, we define a symmetric polynomial $m_{\lambda }\in \mathcal{S}$ by

\[ m_{\lambda }:=\sum _{\substack {a\in \mathbb {N}^{N};\\ \operatorname *{sort}a=\lambda }}x^{a}. \]

This is called the monomial symmetric polynomial corresponding to $\lambda $.

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noncomputable def AlgebraicCombinatorics.SymmetricFunctions.monomialSymm {R : Type u_1} [CommSemiring R] {N : ℕ} (mu : NPartition N) :

MvPolynomial (Fin N) R

AlgebraicCombinatorics.SymmetricFunctions.monomialSymm mu = ∑ a ∈ AlgebraicCombinatorics.SymmetricFunctions.sortPreimage mu, AlgebraicCombinatorics.SymmetricFunctions.monomialExp a

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Proposition 6.63

(a) For each $n\in \left\{ 0,1,\ldots ,N\right\} $, we have

\[ e_{n}=m_{\left( 1,1,\ldots ,1,0,0,\ldots ,0\right) }, \]

where $\left( 1,1,\ldots ,1,0,0,\ldots ,0\right) $ is the $N$-tuple that begins with $n$ many $1$’s and ends with $N-n$ many $0$’s.

(b) For each $n\in \mathbb {N}$, we have

\[ h_{n}=\sum _{\substack {\lambda \text{ is an }N\text{-partition;}\\ \left\vert \lambda \right\vert =n}}m_{\lambda }, \]

where the size $\left\vert \lambda \right\vert $ of an $N$-partition $\lambda $ is defined to be the sum of its entries (i.e., if $\lambda =\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{N}\right) $, then $\left\vert \lambda \right\vert :=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N}$).

(c) Assume that $N{\gt}0$. For each $n\in \mathbb {N}$, we have

\[ p_{n}=m_{\left( n,0,0,\ldots ,0\right) }, \]

where $\left( n,0,0,\ldots ,0\right) $ is the $N$-tuple that begins with an $n$ and ends with $N-1$ zeroes.

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theorem AlgebraicCombinatorics.SymmetricFunctions.elem_symm_eq_monomialSymm {R : Type u_1} [CommSemiring R] {N : ℕ} [DecidableEq (Fin N)] (n : ℕ) (hn : n ≤ N) :

MvPolynomial.esymm (Fin N) R n = monomialSymm (onesThenZeros n hn)

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theorem AlgebraicCombinatorics.SymmetricFunctions.homog_symm_eq_sum_monomialSymm {R : Type u_1} [CommSemiring R] {N : ℕ} [DecidableEq (Fin N)] (n : ℕ) :

MvPolynomial.hsymm (Fin N) R n = ∑ mu ∈ NPartitionsOfSize n, monomialSymm mu

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theorem AlgebraicCombinatorics.SymmetricFunctions.power_sum_eq_monomialSymm {R : Type u_1} [CommSemiring R] {N : ℕ} [DecidableEq (Fin N)] (n : ℕ) (hn : n ≠ 0) (hN : 0 < N) :

MvPolynomial.psum (Fin N) R n = monomialSymm (singletonPartition n hN)

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Theorem 6.78

(a) The family $\left( m_{\lambda }\right) _{\lambda \text{ is an }N\text{-partition}}$ is a basis of the $K$-module $\mathcal{S}$.

(b) Each symmetric polynomial $f\in \mathcal{S}$ satisfies

\[ f=\sum _{\substack {\lambda =\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{N}\right) \\ \text{is an }N\text{-partition}}}\left( \left[ x_{1}^{\lambda _{1}}x_{2}^{\lambda _{2}}\cdots x_{N}^{\lambda _{N}}\right] f\right) m_{\lambda }. \]

(c) Let $n\in \mathbb {N}$. Let

\[ \mathcal{S}_{n}:=\left\{ \text{homogeneous symmetric polynomials }f\in \mathcal{P}\text{ of degree }n\right\} \]

(where we understand the zero polynomial $0\in \mathcal{P}$ to be homogeneous of every degree). Then, $\mathcal{S}_{n}$ is a $K$-submodule of $\mathcal{S}$.

(d) The family $\left( m_{\lambda }\right) _{\lambda \text{ is an }N\text{-partition of size }n}$ is a basis of the $K$-module $\mathcal{S}_{n}$.

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theorem AlgebraicCombinatorics.SymmetricFunctions.monomialSymm_linearIndependent {R : Type u_1} [CommSemiring R] {N : ℕ} (S : Finset (NPartition N)) :

LinearIndependent R fun (mu : ↥S) => monomialSymm ↑mu

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theorem AlgebraicCombinatorics.SymmetricFunctions.monomialSymm_spans {R : Type u_1} [CommSemiring R] {N : ℕ} (f : MvPolynomial (Fin N) R) :

f.IsSymmetric → f ∈ Submodule.span R (Set.range fun (mu : NPartition N) => monomialSymm mu)

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theorem AlgebraicCombinatorics.SymmetricFunctions.symm_eq_sum_coeff_monomialSymm {R : Type u_1} [CommSemiring R] {N : ℕ} (f : MvPolynomial (Fin N) R) (hf : f.IsSymmetric) (S : Finset (NPartition N)) (hS : ∀ (mu : NPartition N), NPartition.size mu ≤ f.totalDegree → mu ∈ S) :

f = ∑ mu ∈ S, MvPolynomial.coeff (Finsupp.equivFunOnFinite.symm mu.parts) f • monomialSymm mu

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def AlgebraicCombinatorics.SymmetricFunctions.symmHomogeneous (N : ℕ) (R : Type u_2) [CommSemiring R] (n : ℕ) :

Submodule R (MvPolynomial (Fin N) R)

AlgebraicCombinatorics.SymmetricFunctions.symmHomogeneous N R n = { carrier := {f : MvPolynomial (Fin N) R | f.IsSymmetric ∧ f.IsHomogeneous n}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

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theorem AlgebraicCombinatorics.SymmetricFunctions.monomialSymm_homogeneous_linearIndependent {R : Type u_1} [CommSemiring R] {N : ℕ} (n : ℕ) :

LinearIndependent R fun (mu : { nu : NPartition N // NPartition.size nu = n }) => monomialSymm ↑mu

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theorem AlgebraicCombinatorics.SymmetricFunctions.monomialSymm_homogeneous_spans {R : Type u_1} [CommSemiring R] {N : ℕ} (n : ℕ) (f : MvPolynomial (Fin N) R) :

f ∈ symmHomogeneous N R n → f ∈ Submodule.span R (Set.range fun (mu : { nu : NPartition N // NPartition.size nu = n }) => monomialSymm ↑mu)

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noncomputable def AlgebraicCombinatorics.SymmetricFunctions.monomialSymm_basis_homogeneous {R : Type u_1} [CommSemiring R] {N : ℕ} (n : ℕ) :

Module.Basis { nu : NPartition N // NPartition.size nu = n } R ↥(symmHomogeneous N R n)

AlgebraicCombinatorics.SymmetricFunctions.monomialSymm_basis_homogeneous n = Module.Basis.mk ⋯ ⋯

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Proposition 6.72

Let $\sigma \in S_{N}$ and $f\in \mathcal{P}$. Then,

\[ \left[ x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{N}^{a_{N}}\right] \left( \sigma \cdot f\right) =\left[ x_{1}^{a_{\sigma \left( 1\right) }}x_{2}^{a_{\sigma \left( 2\right) }}\cdots x_{N}^{a_{\sigma \left( N\right) }}\right] f \]

for any $\left( a_{1},a_{2},\ldots ,a_{N}\right) \in \mathbb {N}^{N}$.

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theorem AlgebraicCombinatorics.SymmetricFunctions.sigma_coeff_permute {R : Type u_1} [CommSemiring R] {N : ℕ} (sigma : Equiv.Perm (Fin N)) (f : MvPolynomial (Fin N) R) (a : Fin N → ℕ) :

MvPolynomial.coeff (Finsupp.equivFunOnFinite.symm a) ((MvPolynomial.rename ⇑sigma) f) = MvPolynomial.coeff (Finsupp.equivFunOnFinite.symm (a ∘ ⇑sigma)) f

Schur polynomials

Alternants

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Definition 6.85

(a) We let $\rho $ be the $N$-tuple $\left( N-1,N-2,\ldots ,N-N\right) \in \mathbb {N}^{N}$.

(b) For any $N$-tuple $\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{N}\right) \in \mathbb {N}^{N}$, we define

\[ a_{\alpha }:=\det \left( \underbrace{\left( x_{i}^{\alpha _{j}}\right) _{1\leq i\leq N,\ 1\leq j\leq N}}_{\in \mathcal{P}^{N\times N}}\right) \in \mathcal{P}. \]

This is called the $\alpha $-alternant (of $x_{1},x_{2},\ldots ,x_{N}$).

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abbrev rhoVector (N : ℕ) :

Fin N → ℕ

rhoVector N = AlgebraicCombinatorics.rho N

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noncomputable abbrev alternant {R : Type u_1} [CommRing R] (N : ℕ) (α : Fin N → ℕ) :

MvPolynomial (Fin N) R

alternant N α = AlgebraicCombinatorics.alternant α

Young diagrams and Schur polynomials

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Definition 6.91

Let $\lambda $ be an $N$-partition.

The Young diagram of $\lambda $ is defined as the set

\[ \left\{ \left( i,j\right) \ \mid \ i\in \left[ N\right] \text{ and }j\in \left[ \lambda _{i}\right] \right\} \subseteq \left\{ 1,2,3,\ldots \right\} ^{2}. \]

We visually represent each element $\left( i,j\right) $ of this Young diagram as a box in row $i$ and column $j$.

We denote the Young diagram of $\lambda $ by $Y\left( \lambda \right) $.

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def NPartition.youngDiagram {N : ℕ} (μ : NPartition N) :

Finset (Fin N × ℕ)

μ.youngDiagram = Finset.univ.biUnion fun (i : Fin N) => Finset.map { toFun := fun (j : ℕ) => (i, j), inj' := ⋯ } (Finset.range (μ.parts i))

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Definition 6.93

Let $\lambda $ be an $N$-partition.

A Young tableau of shape $\lambda $ means a way of filling the boxes of $Y\left( \lambda \right) $ with elements of $\left[ N\right] $ (one element per box). Formally speaking, it is defined as a map $T:Y\left( \lambda \right) \rightarrow \left[ N\right] $.

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structure YoungTableau {N : ℕ} [NeZero N] (lam : NPartition N) :

entry : Fin N × ℕ → Fin N
The filling function
support (c : Fin N × ℕ) : c ∉ lam.youngDiagram → self.entry c = 0
The entry is only meaningful for cells in the diagram

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Definition 6.94

Let $\lambda $ be an $N$-partition.

A Young tableau $T$ of shape $\lambda $ is said to be semistandard if its entries

increase weakly along each row (from left to right);
increase strictly down each column (from top to bottom).

Formally speaking, this means that a Young tableau $T:Y\left( \lambda \right) \rightarrow \left[ N\right] $ is semistandard if and only if

we have $T\left( i,j\right) \leq T\left( i,j+1\right) $ for any $\left( i,j\right) \in Y\left( \lambda \right) $ satisfying $\left( i,j+1\right) \in Y\left( \lambda \right) $;
we have $T\left( i,j\right) {\lt}T\left( i+1,j\right) $ for any $\left( i,j\right) \in Y\left( \lambda \right) $ satisfying $\left( i+1,j\right) \in Y\left( \lambda \right) $.

We let $\operatorname *{SSYT}\left( \lambda \right) $ denote the set of all semistandard Young tableaux of shape $\lambda $.

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structure SSYT {N : ℕ} [NeZero N] (lam : NPartition N) extends YoungTableau lam :

entry : Fin N × ℕ → Fin N
support (c : Fin N × ℕ) : c ∉ lam.youngDiagram → self.entry c = 0
row_weak (i : Fin N) (j₁ j₂ : ℕ) : (i, j₂) ∈ lam.youngDiagram → j₁ < j₂ → self.entry (i, j₁) ≤ self.entry (i, j₂)
Entries increase weakly along rows
col_strict (i₁ i₂ : Fin N) (j : ℕ) : (i₂, j) ∈ lam.youngDiagram → i₁ < i₂ → self.entry (i₁, j) < self.entry (i₂, j)
Entries increase strictly down columns

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Definition 6.101

Let $\lambda $ be an $N$-partition. If $T$ is any Young tableau of shape $\lambda $, then we define the corresponding monomial

\[ x_{T}:=\prod _{c\text{ is a box of }Y\left( \lambda \right) }x_{T\left( c\right) }=\prod _{\left( i,j\right) \in Y\left( \lambda \right) }x_{T\left( i,j\right) }=\prod _{k=1}^{N}x_{k}^{\left( \text{\# of times }k\text{ appears in }T\right) }. \]

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def YoungTableau.monomial {N : ℕ} [NeZero N] {lam : NPartition N} (T : YoungTableau lam) :

T.monomial = ∏ c ∈ lam.youngDiagram, MvPolynomial.X (T.entry c)

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Definition 6.103

Let $\lambda $ be an $N$-partition. We define the Schur polynomial $s_{\lambda }\in \mathcal{P}$ by

\[ s_{\lambda }:=\sum _{T\in \operatorname *{SSYT}\left( \lambda \right) }x_{T}. \]

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def schurPoly {N : ℕ} [NeZero N] (lam : NPartition N) :

schurPoly lam = ∑ f ∈ ssytFillingsYoung lam, fillingMonomialYoung f

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Theorem 6.107

Let $\lambda $ be an $N$-partition. Then:

(a) The polynomial $s_{\lambda }$ is symmetric.

(b) We have

\[ a_{\lambda +\rho }=a_{\rho }\cdot s_{\lambda }. \]

Here, the addition on $\mathbb {N}^{N}$ is defined entrywise.

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theorem schurPoly_isSymmetric {N : ℕ} [NeZero N] (lam : NPartition N) (σ : Equiv.Perm (Fin N)) :

(MvPolynomial.rename ⇑σ) (schurPoly lam) = schurPoly lam

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theorem alternant_eq_rho_mul_schur {N : ℕ} [NeZero N] (lam : NPartition N) :

(alternant N fun (i : Fin N) => lam.parts i + rhoVector N i) = alternant N (rhoVector N) * schurPoly lam

Skew Young diagrams and skew Schur polynomials

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Definition 6.108

Let $\lambda $ and $\mu $ be two $N$-partitions.

We say that $\mu \subseteq \lambda $ if and only if $Y\left( \mu \right) \subseteq Y\left( \lambda \right) $. Equivalently, $\mu \subseteq \lambda $ if and only if

\[ \text{each }i\in \left[ N\right] \text{ satisfies }\mu _{i}\leq \lambda _{i}. \]

Thus we have defined a partial order $\subseteq $ on the set of all $N$-partitions.

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instance NPartition.instLE {N : ℕ} :

LE (NPartition N)

NPartition.instLE = { le := fun (μ ν : NPartition N) => ∀ (i : Fin N), μ.parts i ≤ ν.parts i }

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Definition 6.110

Let $\lambda $ and $\mu $ be two $N$-partitions such that $\mu \subseteq \lambda $. Then, we define the skew Young diagram $Y\left( \lambda /\mu \right) $ to be the set difference

\begin{align*} Y\left( \lambda \right) \setminus Y\left( \mu \right) & =\left\{ \left( i,j\right) \ \mid \ i\in \left[ N\right] \text{ and }j\in \left[ \lambda _{i}\right] \setminus \left[ \mu _{i}\right] \right\} \\ & =\left\{ \left( i,j\right) \ \mid \ i\in \left[ N\right] \text{ and }j\in \mathbb {Z}\text{ and }\mu _{i}{\lt}j\leq \lambda _{i}\right\} . \end{align*}

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def skewYoungDiagram {N : ℕ} [NeZero N] (lam mu : NPartition N) :

Finset (Fin N × ℕ)

skewYoungDiagram lam mu = lam.youngDiagram \ mu.youngDiagram

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Lemma 6.113 Convexity of skew Young diagrams

Let $\lambda $ and $\mu $ be two $N$-partitions such that $\mu \subseteq \lambda $. Let $\left( a,b\right) $ and $\left( e,f\right) $ be two elements of $Y\left( \lambda /\mu \right) $. Let $\left( c,d\right) \in \mathbb {Z}^{2}$ satisfy $a\leq c\leq e$ and $b\leq d\leq f$. Then, $\left( c,d\right) \in Y\left( \lambda /\mu \right) $.

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theorem skewYoungDiagram_convex {N : ℕ} [NeZero N] {lam mu : NPartition N} {a e : Fin N} {b f : ℕ} (hab : (a, b) ∈ skewYoungDiagram lam mu) (hef : (e, f) ∈ skewYoungDiagram lam mu) {c : Fin N} {d : ℕ} (hac : a ≤ c) (hce : c ≤ e) (hbd : b ≤ d) (hdf : d ≤ f) :

(c, d) ∈ skewYoungDiagram lam mu

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Definition 6.114

Let $\lambda $ and $\mu $ be two $N$-partitions such that $\mu \subseteq \lambda $. A Young tableau of shape $\lambda /\mu $ means a way of filling the boxes of $Y\left( \lambda /\mu \right) $ with elements of $\left[ N\right] $ (one element per box). Formally speaking, it is defined as a map $T:Y\left( \lambda /\mu \right) \rightarrow \left[ N\right] $.

Young tableaux of shape $\lambda /\mu $ are often called skew Young tableaux.

If we don’t have $\mu \subseteq \lambda $, then we agree that there are no Young tableaux of shape $\lambda /\mu $.

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structure SkewYoungTableau {N : ℕ} [NeZero N] (lam mu : NPartition N) :

entry : Fin N × ℕ → Fin N
The filling function T : Y(λ/μ) → [N]
support (c : Fin N × ℕ) : c ∉ skewYoungDiagram lam mu → self.entry c = 0
The entry is only meaningful for cells in the skew diagram

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Definition 6.115

Let $\lambda $ and $\mu $ be two $N$-partitions.

A Young tableau $T$ of shape $\lambda /\mu $ is said to be semistandard if its entries

increase weakly along each row (from left to right);
increase strictly down each column (from top to bottom).

Formally speaking, this means that a Young tableau $T:Y\left( \lambda /\mu \right) \rightarrow \left[ N\right] $ is semistandard if and only if

we have $T\left( i,j\right) \leq T\left( i,j+1\right) $ for any $\left( i,j\right) \in Y\left( \lambda /\mu \right) $ satisfying $\left( i,j+1\right) \in Y\left( \lambda /\mu \right) $;
we have $T\left( i,j\right) {\lt}T\left( i+1,j\right) $ for any $\left( i,j\right) \in Y\left( \lambda /\mu \right) $ satisfying $\left( i+1,j\right) \in Y\left( \lambda /\mu \right) $.

We let $\operatorname *{SSYT}\left( \lambda /\mu \right) $ denote the set of all semistandard Young tableaux of shape $\lambda /\mu $.

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structure SkewSSYT {N : ℕ} [NeZero N] (lam mu : NPartition N) extends SkewYoungTableau lam mu :

entry : Fin N × ℕ → Fin N
support (c : Fin N × ℕ) : c ∉ skewYoungDiagram lam mu → self.entry c = 0
row_weak (i : Fin N) (j₁ j₂ : ℕ) : (i, j₁) ∈ skewYoungDiagram lam mu → (i, j₂) ∈ skewYoungDiagram lam mu → j₁ < j₂ → self.entry (i, j₁) ≤ self.entry (i, j₂)
Entries increase weakly along rows
col_strict (i₁ i₂ : Fin N) (j : ℕ) : (i₁, j) ∈ skewYoungDiagram lam mu → (i₂, j) ∈ skewYoungDiagram lam mu → i₁ < i₂ → self.entry (i₁, j) < self.entry (i₂, j)
Entries increase strictly down columns

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Lemma 6.116

Let $\lambda $ and $\mu $ be two $N$-partitions. Let $T$ be a semistandard Young tableau of shape $\lambda /\mu $. Then:

(a) If $\left( i,j_{1}\right) $ and $\left( i,j_{2}\right) $ are two elements of $Y\left( \lambda /\mu \right) $ satisfying $j_{1}\leq j_{2}$, then $T\left( i,j_{1}\right) \leq T\left( i,j_{2}\right) $.

(b) If $\left( i_{1},j\right) $ and $\left( i_{2},j\right) $ are two elements of $Y\left( \lambda /\mu \right) $ satisfying $i_{1}\leq i_{2}$, then $T\left( i_{1},j\right) \leq T\left( i_{2},j\right) $.

(c) If $\left( i_{1},j\right) $ and $\left( i_{2},j\right) $ are two elements of $Y\left( \lambda /\mu \right) $ satisfying $i_{1}{\lt}i_{2}$, then $T\left( i_{1},j\right) {\lt}T\left( i_{2},j\right) $.

(d) If $\left( i_{1},j_{1}\right) $ and $\left( i_{2},j_{2}\right) $ are two elements of $Y\left( \lambda /\mu \right) $ satisfying $i_{1}\leq i_{2}$ and $j_{1}\leq j_{2}$, then $T\left( i_{1},j_{1}\right) \leq T\left( i_{2},j_{2}\right) $.

(e) If $\left( i_{1},j_{1}\right) $ and $\left( i_{2},j_{2}\right) $ are two elements of $Y\left( \lambda /\mu \right) $ satisfying $i_{1}{\lt}i_{2}$ and $j_{1}\leq j_{2}$, then $T\left( i_{1},j_{1}\right) {\lt}T\left( i_{2},j_{2}\right) $.

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theorem SkewSSYT.row_weak_of_le {N : ℕ} [NeZero N] {lam mu : NPartition N} (T : SkewSSYT lam mu) {i : Fin N} {j₁ j₂ : ℕ} (h1 : (i, j₁) ∈ skewYoungDiagram lam mu) (h2 : (i, j₂) ∈ skewYoungDiagram lam mu) (hle : j₁ ≤ j₂) :

T.entry (i, j₁) ≤ T.entry (i, j₂)

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theorem SkewSSYT.col_weak {N : ℕ} [NeZero N] {lam mu : NPartition N} (T : SkewSSYT lam mu) {i₁ i₂ : Fin N} {j : ℕ} (h1 : (i₁, j) ∈ skewYoungDiagram lam mu) (h2 : (i₂, j) ∈ skewYoungDiagram lam mu) (hle : i₁ ≤ i₂) :

T.entry (i₁, j) ≤ T.entry (i₂, j)

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theorem SkewSSYT.col_strict_of_lt {N : ℕ} [NeZero N] {lam mu : NPartition N} (T : SkewSSYT lam mu) {i₁ i₂ : Fin N} {j : ℕ} (h1 : (i₁, j) ∈ skewYoungDiagram lam mu) (h2 : (i₂, j) ∈ skewYoungDiagram lam mu) (hlt : i₁ < i₂) :

T.entry (i₁, j) < T.entry (i₂, j)

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theorem SkewSSYT.monotone {N : ℕ} [NeZero N] {lam mu : NPartition N} (T : SkewSSYT lam mu) {i₁ i₂ : Fin N} {j₁ j₂ : ℕ} (h1 : (i₁, j₁) ∈ skewYoungDiagram lam mu) (h2 : (i₂, j₂) ∈ skewYoungDiagram lam mu) (hi : i₁ ≤ i₂) (hj : j₁ ≤ j₂) :

T.entry (i₁, j₁) ≤ T.entry (i₂, j₂)

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theorem SkewSSYT.strict_monotone {N : ℕ} [NeZero N] {lam mu : NPartition N} (T : SkewSSYT lam mu) {i₁ i₂ : Fin N} {j₁ j₂ : ℕ} (h1 : (i₁, j₁) ∈ skewYoungDiagram lam mu) (h2 : (i₂, j₂) ∈ skewYoungDiagram lam mu) (hi : i₁ < i₂) (hj : j₁ ≤ j₂) :

T.entry (i₁, j₁) < T.entry (i₂, j₂)

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Definition 6.117

Let $\lambda $ and $\mu $ be two $N$-partitions. If $T$ is any Young tableau of shape $\lambda /\mu $, then we define the corresponding monomial

\[ x_{T}:=\prod _{c\text{ is a box of }Y\left( \lambda /\mu \right) }x_{T\left( c\right) }=\prod _{\left( i,j\right) \in Y\left( \lambda /\mu \right) }x_{T\left( i,j\right) }=\prod _{k=1}^{N}x_{k}^{\left( \text{\# of times }k\text{ appears in }T\right) }. \]

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def SkewYoungTableau.monomial {N : ℕ} [NeZero N] {lam mu : NPartition N} (T : SkewYoungTableau lam mu) :

T.monomial = ∏ c ∈ skewYoungDiagram lam mu, MvPolynomial.X (T.entry c)

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Definition 6.125

Let $\lambda $ and $\mu $ be two $N$-partitions. We define the skew Schur polynomial $s_{\lambda /\mu }\in \mathcal{P}$ by

\[ s_{\lambda /\mu }:=\sum _{T\in \operatorname *{SSYT}\left( \lambda /\mu \right) }x_{T}. \]

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def skewSchurPoly {N : ℕ} [NeZero N] (lam mu : NPartition N) :

skewSchurPoly lam mu = ∑ f ∈ ssytFillings lam mu, fillingMonomial f

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Theorem 6.127

Let $\lambda $ and $\mu $ be any two $N$-partitions. Then, the polynomial $s_{\lambda /\mu }$ is symmetric.

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theorem skewSchurPoly_isSymmetric {N : ℕ} [NeZero N] (lam mu : NPartition N) (σ : Equiv.Perm (Fin N)) :

(MvPolynomial.rename ⇑σ) (skewSchurPoly lam mu) = skewSchurPoly lam mu

The Littlewood–Richardson rule

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Definition 6.144

(a) We let $\mathbf{0}$ denote the $N$-tuple $\left( 0,0,\ldots ,0\right) \in \mathbb {N}^{N}$.

(b) Let $\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{N}\right) $ and $\beta =\left( \beta _{1},\beta _{2},\ldots ,\beta _{N}\right) $ be two $N$-tuples in $\mathbb {N}^{N}$. Then, we set

\begin{align*} \alpha +\beta & :=\left( \alpha _{1}+\beta _{1},\alpha _{2}+\beta _{2},\ldots ,\alpha _{N}+\beta _{N}\right) \ \ \ \ \ \ \ \ \ \ \text{and}\\ \alpha -\beta & :=\left( \alpha _{1}-\beta _{1},\alpha _{2}-\beta _{2},\ldots ,\alpha _{N}-\beta _{N}\right) . \end{align*}

Note that $\alpha +\beta \in \mathbb {N}^{N}$, whereas $\alpha -\beta \in \mathbb {Z}^{N}$.

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abbrev AlgebraicCombinatorics.zeroTuple {N : ℕ} :

Fin N → ℕ

AlgebraicCombinatorics.zeroTuple = 0

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abbrev AlgebraicCombinatorics.addTuple {N : ℕ} (α β : Fin N → ℕ) :

Fin N → ℕ

AlgebraicCombinatorics.addTuple α β = α + β

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def AlgebraicCombinatorics.subTuple {N : ℕ} (α β : Fin N → ℕ) :

Fin N → ℤ

AlgebraicCombinatorics.subTuple α β i = ↑(α i) - ↑(β i)

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Definition 6.145

Let $\lambda $ and $\mu $ be two $N$-partitions. Let $T$ be a tableau of shape $\lambda /\mu $. We define the content of $T$ to be the $N$-tuple $\left( a_{1},a_{2},\ldots ,a_{N}\right) $, where

\[ a_{i}:=\left( \text{\# of }i\text{'s in }T\right) =\left( \text{\# of boxes }c\text{ of }T\text{ such that }T\left( c\right) =i\right) . \]

We denote this $N$-tuple by $\operatorname *{cont}T$.

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noncomputable def AlgebraicCombinatorics.contentTableau {N : ℕ} {lam mu : Fin N → ℕ} (T : Tableau lam mu) :

Fin N → ℕ

AlgebraicCombinatorics.contentTableau T i = Nat.card { c : { c : Fin N × ℕ // c ∈ AlgebraicCombinatorics.skewYoungDiagram lam mu } // T c = i }

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Definition 6.146

Let $\lambda $ and $\mu $ be two $N$-partitions. Let $T$ be a tableau of shape $\lambda /\mu $. Let $j$ be a positive integer. Then, $\operatorname {col}_{\geq j}T$ means the restriction of $T$ to columns $j,j+1,j+2,\ldots $ (that is, the result of removing the first $j-1$ columns from $T$). Formally speaking, this means the restriction of the map $T$ to the set $\left\{ \left( u,v\right) \in Y\left( \lambda /\mu \right) \ \mid \ v\geq j\right\} $.

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def AlgebraicCombinatorics.colGeq {N : ℕ} {lam mu : Fin N → ℕ} (T : Tableau lam mu) (j : ℕ) :

{ c : Fin N × ℕ // c ∈ skewYoungDiagram lam mu ∧ c.2 ≥ j } → Fin N

AlgebraicCombinatorics.colGeq T j ⟨c, hc⟩ = T ⟨c, ⋯⟩

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Definition 6.147

Let $\lambda ,\mu ,\nu $ be three $N$-partitions. A semistandard tableau $T$ of shape $\lambda /\mu $ is said to be $\nu $-Yamanouchi (this is an adjective) if for each positive integer $j$, the $N$-tuple $\nu +\operatorname *{cont}\left( \operatorname {col}_{\geq j}T\right) \in \mathbb {N}^{N}$ is an $N$-partition (i.e., weakly decreasing).

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def AlgebraicCombinatorics.IsYamanouchi {N : ℕ} {lam mu : Fin N → ℕ} (nu : Fin N → ℕ) (T : Tableau lam mu) :

AlgebraicCombinatorics.IsYamanouchi nu T = (AlgebraicCombinatorics.IsSemistandard T ∧ ∀ j > 0, AlgebraicCombinatorics.IsNPartition (nu + AlgebraicCombinatorics.contentColGeq T j))

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Theorem 6.196 Zelevinsky’s generalized Littlewood–Richardson rule, in Yamanouchi form

Let $\lambda ,\mu ,\nu $ be three $N$-partitions. Then,

\begin{equation} s_{\nu }\cdot s_{\lambda /\mu }=\sum _{\substack {T\text{ is a }\nu \text{-Yamanouchi}\\ \text{semistandard tableau}\\ \text{of shape }\lambda /\mu }}s_{\nu +\operatorname *{cont}T}. \label{eq.thm.sf.lr-zy.eq}\end{equation}

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theorem AlgebraicCombinatorics.littlewoodRichardson {R : Type u_1} [CommRing R] {N : ℕ} [IsDomain R] (lam mu nu : Fin N → ℕ) (hlam : IsNPartition lam) (hmu : IsNPartition mu) (hnu : IsNPartition nu) :

schurPoly nu * skewSchurPoly lam mu = ∑ T : { T : Tableau lam mu // IsYamanouchi nu T }, schurPoly (nu + contentTableau ↑T)

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Lemma 6.188 Stembridge’s Lemma

Let $\lambda ,\mu ,\nu $ be three $N$-partitions. Then,

\[ a_{\nu +\rho }\cdot s_{\lambda /\mu }=\sum _{\substack {T\text{ is a }\nu \text{-Yamanouchi}\\ \text{semistandard tableau}\\ \text{of shape }\lambda /\mu }}a_{\nu +\operatorname *{cont}T+\rho }. \]

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theorem AlgebraicCombinatorics.stembridgeLemma {R : Type u_1} [CommRing R] {N : ℕ} (lam mu nu : Fin N → ℕ) (hlam : IsNPartition lam) (hmu : IsNPartition mu) (hnu : IsNPartition nu) :

alternant (nu + rho N) * skewSchurPoly lam mu = ∑ T : { T : Tableau lam mu // IsYamanouchi nu T }, alternant (nu + contentTableau ↑T + rho N)

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Lemma 6.189

Let $\lambda $ be any $N$-partition. Let $T$ be a semistandard tableau of shape $\lambda $. Then, $T\left( i,j\right) \geq i$ for each $\left( i,j\right) \in Y\left( \lambda \right) $.

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theorem AlgebraicCombinatorics.tableau_entry_ge_row {N : ℕ} {lam : Fin N → ℕ} (hlam : IsNPartition lam) (T : Tableau lam zeroTuple) (hT : IsSemistandard T) (c : { c : Fin N × ℕ // c ∈ skewYoungDiagram lam zeroTuple }) :

(↑c).1 ≤ T c

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Definition 6.153

Let $L$ be a commutative ring. Let $a\in L$. The element $a$ of $L$ is said to be regular if and only if every $x\in L$ satisfying $ax=0$ satisfies $x=0$.

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def AlgebraicCombinatorics.IsRegularElement {L : Type u_2} [CommRing L] (a : L) :

AlgebraicCombinatorics.IsRegularElement a = ∀ (x : L), a * x = 0 → x = 0

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Lemma 6.154

Let $L$ be a commutative ring. Let $a,u,v\in L$ be such that $a$ is regular. Assume that $au=av$. Then, $u=v$.

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theorem AlgebraicCombinatorics.IsRegularElement.cancel {L : Type u_2} [CommRing L] {a u v : L} (ha : IsRegularElement a) (h : a * u = a * v) :

u = v

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Lemma 6.155

The element $a_{\rho }$ of the polynomial ring $\mathcal{P}$ is regular.

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theorem AlgebraicCombinatorics.alternant_rho_isRegular {R : Type u_1} [CommRing R] {N : ℕ} [IsDomain R] :

IsRegular (alternant (rho N))

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Lemma 6.90

Let $\alpha \in \mathbb {N}^{N}$.

(a) If the $N$-tuple $\alpha $ has two equal entries, then $a_{\alpha }=0$.

(b) Let $\beta \in \mathbb {N}^{N}$ be an $N$-tuple obtained from $\alpha $ by swapping two entries. Then, $a_{\beta }=-a_{\alpha }$.

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theorem alternant_zero_of_eq (N : ℕ) {R : Type u_2} [CommRing R] {α : Fin N → ℕ} (i j : Fin N) (hij : i ≠ j) (heq : α i = α j) :

alternant N α = 0

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theorem alternant_swap (N : ℕ) {R : Type u_2} [CommRing R] {α : Fin N → ℕ} (i j : Fin N) (hij : i ≠ j) :

alternant N (α ∘ ⇑(Equiv.swap i j)) = -alternant N α

The Pieri rules

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Definition 6.197

Let $\lambda $ and $\mu $ be two $N$-partitions.

(a) We write $\lambda /\mu $ for the pair $\left( \mu ,\lambda \right) $. Such a pair is called a skew partition.

(b) We say that $\lambda /\mu $ is a horizontal strip if we have $\mu \subseteq \lambda $ and the Young diagram $Y\left( \lambda /\mu \right) $ has no two boxes lying in the same column.

(c) We say that $\lambda /\mu $ is a vertical strip if we have $\mu \subseteq \lambda $ and the Young diagram $Y\left( \lambda /\mu \right) $ has no two boxes lying in the same row.

Now, let $n\in \mathbb {N}$.

(d) We say that $\lambda /\mu $ is a horizontal $n$-strip if $\lambda /\mu $ is a horizontal strip and satisfies $\left\vert Y\left( \lambda /\mu \right) \right\vert =n$.

(e) We say that $\lambda /\mu $ is a vertical $n$-strip if $\lambda /\mu $ is a vertical strip and satisfies $\left\vert Y\left( \lambda /\mu \right) \right\vert =n$.

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structure SymmetricFunctions.SkewPartition (N : ℕ) :

outer : NPartition N
The outer partition λ
inner : NPartition N
The inner partition μ
contained : self.inner ≤ self.outer
The containment condition μ ⊆ λ

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def SymmetricFunctions.SkewPartition.isHorizontalStrip {N : ℕ} (s : SkewPartition N) :

s.isHorizontalStrip = ∀ (i : Fin N) (hi : ↑i + 1 < N), s.inner.parts i ≥ s.outer.parts ⟨↑i + 1, hi⟩

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def SymmetricFunctions.SkewPartition.isVerticalStrip {N : ℕ} (s : SkewPartition N) :

s.isVerticalStrip = ∀ (i : Fin N), s.outer.parts i ≤ s.inner.parts i + 1

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def SymmetricFunctions.SkewPartition.isHorizontalNStrip {N : ℕ} (s : SkewPartition N) (n : ℕ) :

s.isHorizontalNStrip n = (s.isHorizontalStrip ∧ s.size = n)

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def SymmetricFunctions.SkewPartition.isVerticalNStrip {N : ℕ} (s : SkewPartition N) (n : ℕ) :