More refined enumerations of alternating sign matrices

TL;DR

This paper advances the enumeration of alternating sign matrices by refining counts based on multiple initial rows, deriving linear equations for these refined counts, and proposing conjectural formulas and extensions.

Contribution

It introduces a new refinement of ASM enumeration considering multiple rows, derives linear equations for these counts, and presents conjectures for explicit formulas and generalizations.

Findings

Derived linear equations for doubly-refined enumeration numbers

Proposed a conjectural explicit formula for enumeration numbers

Formulated conjectures on the sufficiency of equations and extensions

Abstract

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general "d-refined" enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_{n,i,j} that enumerate such matrices. We give a conjectural explicit formula for A_{n,i,j} and formulate several other conjectures about the sufficiency of the linear equations to determine the A_{n,i,j}'s and about an extension of the linear equations to the general d-refined enumerations.