Integrability and combinatorics

TL;DR

This paper explores how integrable systems techniques can be applied to solve problems in enumerative and algebraic combinatorics, illustrating with examples like Alternating Sign Matrices and symmetric polynomials.

Contribution

It introduces new methods from integrable systems to address combinatorial enumeration and algebraic problems, demonstrating their effectiveness through specific examples.

Findings

Enumeration formulas for Alternating Sign Matrices derived

Connections established between integrable systems and symmetric polynomials

New combinatorial identities proposed

Abstract

We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the theory of symmetric polynomials.