On refined enumerations of some symmetry classes of ASMs

TL;DR

This paper derives explicit formulas for refined counts of certain symmetry classes of alternating-sign matrices using determinant representations and confirms the Kutin-Yuen conjecture.

Contribution

It provides new explicit formulas for refined enumerations of symmetry classes of ASMs and proves the Kutin-Yuen conjecture.

Findings

Explicit formulas for refined enumerations of symmetry classes of ASMs

Proof of the Kutin-Yuen conjecture

Application of determinant representations and the recent method

Abstract

Using determinant representations for partition functions of the corresponding square ice models and the method proposed recently by one of the authors, we investigate refined enumerations of vertically symmetric alternating-sign matrices, off-diagonally symmetric alternating-sign matrices and alternating-sign matrices with U-turn boundary. For all these cases the explicit formulas for refined enumerations are found. It particular, Kutin-Yuen conjecture is proved.