Enumeration of quarter-turn symmetric alternating-sign matrices of odd order

TL;DR

This paper introduces a new square-ice model for odd-order quarter-turn symmetric alternating-sign matrices, enabling the proof of Robbins' conjectures on their enumeration by expressing the partition function as a product of two factors.

Contribution

It extends Kuperberg's approach to odd-order matrices, providing a novel model and a proof of enumeration conjectures for quarter-turn symmetric alternating-sign matrices.

Findings

Partition function expressed as a product of two factors

Proof of Robbins' enumeration conjectures

Bijection between model states and matrices

Abstract

It was shown by Kuperberg that the partition function of the square-ice model related to the quarter-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with the quarter-turn symmetric alternating-sign matrices of odd order, and show that the partition function of this model can be also written in a similar way. This allows to prove, in particular, the conjectures by Robbins related to the enumeration of the quarter-turn symmetric alternating-sign matrices.