Algebraic Combinatorics

This is the root import file for the Algebraic Combinatorics library. Importing this file provides access to all public-facing modules in the library.

Namespace Convention

This library uses two top-level namespaces:

• AlgebraicCombinatorics — the primary namespace for most definitions and theorems
• AlgComb — a shorter alias used in some modules (determinants, symmetric functions)

Both namespaces are valid and coexist intentionally. When writing new code:

• Use AlgebraicCombinatorics for new files unless extending existing AlgComb modules
• Use AlgComb when adding to files that already use this namespace
• Cross-namespace references work via explicit qualification or open declarations

The two namespaces arose organically during development. AlgComb is shorter and convenient for frequently-used definitions, while AlgebraicCombinatorics is more descriptive.

Contents

The library is organized into the following major areas:

Formal Power Series (FPS)

Basic definitions, operations, and properties of formal power series.

Partitions

Integer partitions, q-binomial coefficients, and related combinatorics.

Permutations

Permutation groups, inversions, signs, cycles, and Lehmer codes.

Signed Counting

Inclusion-exclusion, alternating sums, and Möbius inversion.

Symmetric Functions

Symmetric polynomials, monomial symmetric functions, Schur polynomials, and the Littlewood-Richardson rule.

Determinants

Basic properties, Cauchy-Binet formula, Desnanot-Jacobi identity, and the Lindström-Gessel-Viennot lemma.