(a) (\(\Rightarrow \)): A horizontal strip satisfies \(\mu _i \geq \lambda _{i+1}\) for all valid \(i\) by definition, and \(\lambda _i \geq \mu _i\) follows from the containment condition \(\mu \subseteq \lambda \). (\(\Leftarrow \)): The interlacing condition \(\mu _i \geq \lambda _{i+1}\) is exactly the horizontal strip condition.

(b) (\(\Rightarrow \)): A vertical strip satisfies \(\lambda _i \leq \mu _i + 1\) by definition, and \(\mu _i \leq \lambda _i\) follows from containment. (\(\Leftarrow \)): The condition \(\lambda _i \leq \mu _i + 1\) is exactly the vertical strip condition.

Immediate from the definitions.

Immediate from the definitions.

Combine the horizontal \(n\)-strip hypothesis with the size-difference computation and apply the membership characterization.

Combine the vertical \(n\)-strip hypothesis with the size-difference computation and apply the membership characterization.

The bijection maps \((i, j)\) to \((j, i)\). The transpose condition \(\lambda _i {\gt} j \iff \lambda ^t_j {\gt} i\) guarantees that this map is well-defined and bijective.

Both sums count the cardinality of the respective Young diagram box sets. The bijection from Lemma 6.205 shows these sets have the same cardinality.

The only SSYT of shape \(0\) is the empty tableau, whose monomial is \(1\).

A skew partition \(\lambda /0\) is just the partition \(\lambda \), so \(\mathrm{SSYT}(\lambda )\) and \(\mathrm{SSYT}(\lambda /0)\) are in natural bijection preserving monomials.

This is the reverse direction of Lemma 6.210.

Both sides are sums of monomials: \(h_n\) sums over multisets of size \(n\) from \(\mathrm{Fin}\, N\), while \(s_{(n,0,\ldots ,0)}\) sums over SSYT of row partition shape. There is a natural bijection identifying a multiset of size \(n\) with the unique SSYT whose only nonempty row (row \(0\)) contains the sorted multiset entries. Weight preservation follows from the fact that both sides compute the product of \(x_j\) over the entries.

Both sides are sums of monomials: \(e_n\) sums over subsets of \(\mathrm{Fin}\, N\) of size \(n\), while \(s_{(1^n,0^{N-n})}\) sums over SSYT of column partition shape (single-column tableaux with \(n\) rows). There is a natural bijection identifying a size-\(n\) subset with the SSYT whose column entries are the sorted elements of the subset. Injectivity, surjectivity, and weight preservation are verified explicitly.

When all entries are \(0\), the content \(\mathrm{cont}_{\geq j}(T)\) at any position \(k {\gt} 0\) is \(0\) (no cell has entry \(k\)). Therefore \(\nu + \mathrm{cont}_{\geq j}(T) = (n + c_0, 0, \ldots , 0)\) for some \(c_0 \geq 0\), which is weakly decreasing.

Sorting a multiset and extracting it gives back the original multiset.

By induction on \(k_2 - k_1\): the consecutive case uses column-strictness of SSYT directly; the general case follows by transitivity.

Unfold the filter conditions and relate them to the strip definitions.

Unfold the filter conditions and relate them to the strip definitions.

(a) The proof follows from the Littlewood–Richardson rule (Theorem 6.196) by specializing to the case where one of the partitions is a single row \((n, 0, 0, \ldots , 0)\).

Alternatively, a direct combinatorial proof uses the RSK row insertion:

1. Expand \(h_n = \sum _{s \in \mathrm{Sym}(\mathrm{Fin}\, N, n)} \prod _{j \in s} x_j\) and \(s_\mu = \sum _{T \in \mathrm{SSYT}(\mu )} x_T\).

2. Distribute to get \(h_n \cdot s_\mu = \sum _{s, T} \prod _{j \in s} x_j \cdot x_T\).

3. Apply the RSK row insertion bijection: \(\{ (s, T) : \mathrm{Sym}(\mathrm{Fin}\, N, n) \times \mathrm{SSYT}(\mu )\} \simeq \bigsqcup _{\lambda /\mu \text{ horiz } n\text{-strip}} \mathrm{SSYT}(\lambda )\). Given a sorted multiset \(s = \{ a_1, \ldots , a_n\} \) and tableau \(T\) of shape \(\mu \), row-insert \(a_1, \ldots , a_n\) into \(T\) to get a tableau \(T'\) of shape \(\lambda \) where \(\lambda /\mu \) is a horizontal \(n\)-strip.

4. Weight preservation: \(\prod _{j \in s} x_j \cdot x_T = x_{T'}\).

5. Conclude \(h_n \cdot s_\mu = \sum _\lambda s_\lambda \).

(b) Analogous using RSK column insertion, or via the \(\omega \)-involution.

Roundtrip properties follow from the fact that sorting a weakly increasing list is the identity, and sorting a multiset then extracting gives back the multiset.

When \(c - a \geq 0\), the bijection between lattice paths and \(\mathrm{Sym}(\mathrm{Fin}\, N, (c-a))\) (via Lemma 6.229) gives:

When \(c - a {\lt} 0\), both sides are \(0\).

The entries of each row of the constructed tableau are the east-step heights of the original nipat. Converting back produces the same east-step lists.

The proof uses entry specification lemmas to show the entries agree pointwise.

The weight of a nipat is \(\prod _i w(p_i)\) and the monomial of the SSYT is \(\prod _i \prod _k x_{T(i,k)}\). Since the \(k\)-th entry of row \(i\) in the tableau equals the \(k\)-th east-step height of path \(p_i\), these are equal.

Combine the two roundtrip lemmas.

Transport the sum along the equivalence with weight preservation.

The forward map sends a nipat \(\mathbf{p}\) to the tableau \(T(\mathbf{p})\); the inverse extracts the east-step heights from each row of a skew SSYT. The bijectivity is proved via:

• Injectivity: Lemma 6.234 shows that converting a nipat to a tableau and back gives the original nipat.

• Surjectivity: Lemma 6.235 shows that converting a tableau to a nipat and back gives the original tableau.

Weight preservation (Lemma 6.236) follows from the fact that the weight of each component path is the product of \(x_j\) over its east-step heights, which are exactly the row entries of the tableau.

Follows from the nipat–SSYT bijection (Theorem 6.239) and the definition of \(s_{\lambda /\mu }\) as a sum over SSYT monomials. The proof uses Lemma 6.238 followed by a bijective reindexing to relate the finset sum to the type sum.

Direct computation from the weakly decreasing property of partitions.

Analogous to Lemma 6.243.

By induction on the vertex list: each east-step increases the count by \(1\) and the \(x\)-coordinate by \(1\); north-steps change neither.

By induction on the vertex list. For an east-step from \(v_0\) to \(v_1\), the \(y\)-coordinates satisfy \(y(v_0) = y(v_1)\), and the remaining heights are \(\geq y(v_1)\) by the inductive hypothesis and \(y\)-monotonicity.

Each east-step \(y\)-coordinate is the \(y\)-coordinate of some vertex of the path. The bound follows from \(y\)-monotonicity along the path.

For \(c - a \geq 0\): compose the LGV-path-to-lattice-path bijection with the lattice path–multiset equivalence (Lemma 6.229), then apply Theorem 6.231. For \(c - a {\lt} 0\): the set of LGV paths is empty (no path can have decreasing \(x\)-coordinate), and \(h_{c-a} = 0\) by convention.

Reduce to showing that the product of \(x_{y-1}\) over east-step \(y\)-coordinates equals the product of \(x_h\) over the converted \(\mathrm{Fin}\, N\) heights.

Two paths with the same start, end, and east-step \(y\)-coordinates must have identical vertex lists, since at each step the direction (north or east) is determined by the remaining east-step \(y\)-coordinates and the current position.

Each entry \(H_{i,j} = h_{\lambda _i - \mu _j - i + j}\) equals the path weight sum from \(A_j\) to \(B_i\) by Theorem 6.249, since the path length is \((\lambda _i - i) - (\mu _j - j) = \lambda _i - \mu _j - i + j\). The transposition accounts for the index swap.

Apply \(\det (H) = \det (M^T) = \det (M)\) using Lemma 6.252 and the transpose-determinant identity.

This requires constructing an explicit weight-preserving bijection between LGV path tuples and nipats. The bijection extracts east-step \(y\)-coordinates from LGV paths and converts them to weakly increasing sequences in \(\mathrm{Fin}\, N\). Non-intersection in the LGV sense corresponds to column-strictness in the nipat sense. Injectivity follows from Lemma 6.251; weight preservation from Lemma 6.250.

By the integer lattice topology: if \(p\) is above \(p'\) at column \(x_0\) but \(p'\) rises above \(p\) at some later column, then by the intermediate value property of lattice paths, the paths must share a vertex, contradicting non-intersection.

The proof proceeds in steps:

1. \(\det (H) = \det (M)\) by Lemma 6.253.

2. Apply the LGV nonpermutable lemma to obtain \(\det (M) = \sum _{\text{LGV-nipats}} w(\mathbf{p})\). The required sorting conditions on \(\mathbf{A}\) and \(\mathbf{B}\) follow from Lemmas 6.243 and 6.244.

3. Connect LGV non-intersecting path tuples to non-intersecting path tuples (Definition 6.230) via Lemma 6.254.

The proof combines two key lemmas. Lemma 6.240 yields

and then Lemma 6.256 gives

From the first Jacobi–Trudi formula (Theorem 6.257) applied to the transpose partitions \(\lambda ^t\) and \(\mu ^t\): \(s_{\lambda ^t/\mu ^t} = \det (h_{(\lambda ^t)_i - (\mu ^t)_j - i + j})\). Applying the \(\omega \)-involution (\(\omega (h_n) = e_n\), \(\omega (s_{\lambda /\mu }) = s_{\lambda ^t/\mu ^t}\)) and using that \(\omega ^2 = \mathrm{id}\) gives \(s_{\lambda /\mu } = \omega (s_{\lambda ^t/\mu ^t}) = \det (\omega (h_{(\lambda ^t)_i - (\mu ^t)_j - i + j})) = \det (e_{(\lambda ^t)_i - (\mu ^t)_j - i + j})\).

Relate \(s_\lambda \) to \(s_{\lambda /0}\) via Lemma 6.211, then apply the general Jacobi–Trudi formula (Theorem 6.257) with \(\mu = 0\).

By case analysis on \(n\): the non-negative case uses the symmetry of \(h_n\); the negative case is trivial.

For any permutation \(\sigma \) of \(\mathrm{Fin}\, N\), the renaming by \(\sigma \) commutes with determinant (as a ring homomorphism preserves sums, products, and signs). Since each entry is fixed by the renaming, the determinant is also fixed.

Each entry is an extended \(h\)-polynomial, which is symmetric.

Apply Lemma 6.261 with the entry symmetry from Lemma 6.262.

By Theorem 6.257, \(s_{\lambda /\mu } = \det (H)\). By Theorem 6.263, \(\det (H)\) is symmetric.

By Lemma 6.211, \(s_\lambda = s_{\lambda /0}\). Apply Theorem 6.264.

By definition of \(\omega \): since \(\omega \) is the composition of the \(h\)-algebra homomorphism with the inverse of the \(e\)-algebra equivalence, applying it to \(e_{k+1}\) first sends \(e_{k+1}\) to the generator \(X_k\), then maps \(X_k\) to \(h_{k+1}\).

Reindex the standard Newton–Girard identity \(\sum _j (-1)^j e_j h_{n-j} = 0\) via \(j \mapsto n - j\) to obtain \(\sum _j (-1)^{n-j} h_{n-j} e_j = 0\). Then factor out \((-1)^n\) and use the sign identity \((-1)^{n-j} = (-1)^n \cdot (-1)^j\) (up to rearrangement and \((-1)^{2j} = 1\)).

From \(\sum _{j=0}^n (-1)^j e_j h_{n-j} = 0\), isolate \(j = 0\): \(h_n + \sum _{j=1}^n (-1)^j e_j h_{n-j} = 0\), hence \(h_n = -\sum _{j=1}^n (-1)^j e_j h_{n-j} = \sum _{j=1}^n (-1)^{j+1} e_j h_{n-j}\).

Analogous to Lemma 6.270.

By strong induction on \(k\):

• Base case (\(k = 0\)): \(h_1 = e_1\), so \(\omega (h_1) = \omega (e_1) = h_1 = e_1\).

• Inductive step: Use the Newton–Girard recurrence (Lemma 6.270) to write \(h_{k+1} = \sum _{j=1}^{k+1} (-1)^{j+1}\, e_j\, h_{k+1-j}\). Apply \(\omega \): by Theorem 6.268, \(\omega (e_j) = h_j\), and by the IH, \(\omega (h_{k+1-j}) = e_{k+1-j}\) for \(k+1-j \leq k\). Hence \(\omega (h_{k+1}) = \sum _{j=1}^{k+1} (-1)^{j+1}\, h_j\, e_{k+1-j}\), which equals \(e_{k+1}\) by the symmetric Newton–Girard recurrence (Lemma 6.271).