A formula for the doubly-refined enumeration of alternating sign matrices

TL;DR

This paper derives an explicit formula for the number of alternating sign matrices with specified top two rows, advancing the understanding of doubly-refined enumeration beyond previous partial results.

Contribution

It provides a complete explicit formula for the doubly-refined enumeration of alternating sign matrices with given top two rows, building on Stroganov's earlier work.

Findings

Derived an explicit formula for doubly-refined enumeration

Extended previous partial results to a full formula

Enhanced understanding of alternating sign matrices with fixed top two rows

Abstract

Zeilberger proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey, for the number of alternating sign matrices with given top row. Stroganov proved an explicit formula for the number of alternating sign matrices with given top and bottom rows. Fischer and Romik considered a different kind of "doubly-refined enumeration" where one counts alternating sign matrices with given top two rows, and obtained partial results on this enumeration. In this paper we continue the study of the doubly-refined enumeration with respect to the top two rows, and use Stroganov's formula to prove an explicit formula for these doubly-refined enumeration numbers.