Enumeration of alternating sign matrices of even size (quasi-)nvariant under a quarter-turn rotation

TL;DR

This paper derives an enumeration formula for alternating sign matrices of even size that are quasi-invariant under a quarter-turn rotation, linking counts of various symmetric ASM classes.

Contribution

It provides a proof for Duchon's conjectured enumeration formula involving quasi-invariant ASM counts.

Findings

Derived an explicit enumeration formula for quasi-invariant ASM under quarter-turn.

Connected counts of quasi-invariant ASM with unrestricted and half-turn symmetric ASM.

Confirmed Duchon's conjecture on ASM enumeration under specific symmetry conditions.

Abstract

The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.