1/N phenomenon for some symmetry classes of the odd alternating sign matrices

TL;DR

This paper studies odd-order alternating sign matrices with central symmetry, proposing conjectures about simple ratios of their counts, extending known results for half-turn symmetric cases to other symmetry classes.

Contribution

It introduces conjectures on enumeration ratios for symmetric odd-order alternating sign matrices, generalizing recent findings for half-turn symmetry.

Findings

Established ratio for half-turn symmetric matrices: (m+1)/m

Proposed conjectures for quarter-turn and diagonal flip symmetries

Identified structural properties of centrally symmetric matrices

Abstract

We consider the alternating sign matrices of the odd order that have some kind of central symmetry. Namely, we deal with matrices invariant under the half-turn, quarter-turn and flips in both diagonals. In all these cases, there are two natural structures in the centre of the matrix. For example, for the matrices invariant under the half-turn the central element is equal $\pm 1$. It was recently found that $A^+_{HT}(2m+1)/A^-_{HT}(2m+1)$=(m+1)/m. We conjecture that similar very simple relations are valid in the two remaining cases.