Yet another doubly refined enumeration of Alternating Sign Matrices

TL;DR

This paper introduces a new determinantal formula for the doubly refined enumeration of alternating sign matrices with three parameters, advancing the combinatorial understanding of these matrices and refining existing conjectures.

Contribution

It derives a novel determinantal formula for the doubly refined enumeration of alternating sign matrices with three parameters, building on symmetry functions and previous conjectures.

Findings

New determinantal formula for doubly refined enumeration

Refinement of the decomposition conjecture by Mills et al.

Enhanced understanding of parameters in alternating sign matrices

Abstract

Since the alternating sign matrix conjecture, proposed by Mills, Robbins, and Rumsey in 1982, was proved by Zeilberger and Kuperberg, several refined enumerations have been considered. In particular, Behrend et al. obtained a quadruply refined enumeration by adding certain parameters. In this paper, we revisit the doubly refined enumeration of alternating sign matrices by adding three parameters: the number of $-1$'s, the position of the $1$ in the first row, and the position of the $1$ in the last row. Using Lascoux's formula on symmetry functions, we derive a new determinantal formula for this doubly refined enumeration. Besides the enumeration conjecture, Mills et al. also proposed a decomposition conjecture, which was subsequently proven by Kuperberg. We present a refinement of that decomposition conjecture.