New enumeration formulas for alternating sign matrices and square ice partition functions

TL;DR

This paper derives new explicit formulas for enumerating alternating sign matrices with refined boundary conditions, connecting these to partition functions of the square ice model at a specific parameter, advancing combinatorial enumeration methods.

Contribution

It introduces novel enumeration formulas for triply- and quadruply-refined ASMs, extending previous results and providing new representations for the square ice model's partition function.

Findings

Explicit formulas for triply- and quadruply-refined ASM enumeration.

New representations for the square ice model partition function at the combinatorial point.

Solution to the full boundary correlation function problem for ASMs.

Abstract

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and triply-refined enumeration by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the "top-left-bottom" (triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the "combinatorial point" 2{\pi}/3.