Linear relations of refined enumerations of alternating sign matrices

TL;DR

This paper establishes a linear relation between refined counts of alternating sign matrices with fixed top, bottom, left, and right columns, suggesting a reduction to simpler enumeration cases.

Contribution

It introduces a linear relation connecting different refined enumerations of alternating sign matrices with fixed boundary rows and columns.

Findings

Derived a simple linear relation between two classes of refined enumerations.

Proposed a system of linear equations to determine refined counts uniquely.

Indicated potential reduction of complex boundary conditions to simpler fixed-row enumerations.

Abstract

In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition a number of left and right columns are fixed. The main result is a simple linear relation between the number of $n \times n$ alternating sign matrices where the top row as well as the left and the right column is fixed and the number of $n \times n$ alternating sign matrices where the two top rows and the bottom row is fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only a number of top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.