Diagonally and antidiagonally symmetric alternating sign matrices of odd order

TL;DR

This paper proves a long-standing conjecture by deriving a formula for counting diagonally and antidiagonally symmetric alternating sign matrices of odd order using a six-vertex model and Schur functions.

Contribution

It introduces a novel six-vertex model approach and proves two conjectures related to the enumeration and ratio of DASASMs of odd order.

Findings

The total number of (2n+1)×(2n+1) DASASMs is given by a specific product formula.

The ratio of DASASMs with central entries -1 and 1 is n/(n+1).

The enumeration formula for odd-order DASASMs is now fully established.

Abstract

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of $(2n+1)\times(2n+1)$ DASASMs is $\prod_{i=0}^n\frac{(3i)!}{(n+i)!}$, and a conjecture of Stroganov from 2008 that the ratio between the numbers of $(2n+1)\times(2n+1)$ DASASMs with central entry $-1$ and $1$ is $n/(n+1)$. Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.