# Refined Enumeration of Symmetry Classes of Alternating Sign Matrices

## TL;DR

This paper provides refined enumeration formulas for various symmetry classes of alternating sign matrices, confirming several conjectures using the six-vertex model from statistical physics.

## Contribution

It introduces new enumeration results for multiple symmetry classes of alternating sign matrices, validating conjectures by Fischer, Duchon, and Robbins.

## Key findings

- Enumeration formulas for symmetry classes of ASMs
- Confirmation of conjectures on ASM symmetry classes
- Application of the six-vertex model to enumeration

## Abstract

We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.