Enumerations of half-turn symmetric alternating-sign matrices of odd order

TL;DR

This paper introduces a new square-ice model for odd-order half-turn symmetric alternating-sign matrices, deriving their enumeration and partition function expressions, extending previous even-order results.

Contribution

It proposes a novel square-ice model for odd-order matrices and derives explicit enumeration formulas, expanding the combinatorial understanding of these matrices.

Findings

Partition function expressed via known factors.

Enumeration of matrices with central entry 1 or -1.

Extension of Kuperberg's even-order results to odd order.

Abstract

It was shown by Kuperberg that the partition function of the square-ice model related to half-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 half-turn symmetric alternating-sign matrices of odd order. The partition function of the model is expressed via the above mentioned factors. The contributions to the partition function of the states corresponding to the alternating-sign matrices having 1 or -1 as the central entry are found and the related enumerations are obtained.