Multiply-refined enumeration of alternating sign matrices

TL;DR

This paper develops a recursive quadratic relation for generating functions of six boundary and bulk statistics in alternating sign matrices, advancing enumeration techniques through determinant identities and integrable model formulas.

Contribution

It introduces a new quadratic relation that recursively determines the generating function for six ASM statistics, connecting combinatorics with integrable models.

Findings

Derived a quadratic recursive relation for ASM statistics

Established new identities for generating functions with fewer statistics

Connected ASM enumeration with the Izergin-Korepin formula

Abstract

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.