Diagonally symmetric alternating sign matrices

TL;DR

This paper derives a Pfaffian formula for counting diagonally symmetric alternating sign matrices (DSASMs), providing the first exact enumeration for this symmetry class where simple product formulas are absent, and explores related generating functions and conjectures.

Contribution

It introduces a Pfaffian formula for DSASMs and their generating functions, advancing enumeration methods for symmetry classes lacking simple formulas.

Findings

Pfaffian formula for DSASM enumeration

Generating functions for DSASM statistics derived

Conjectures on asymptotic enumeration and related classes

Abstract

The enumeration of diagonally symmetric alternating sign matrices (DSASMs) is studied, and a Pfaffian formula is obtained for the number of DSASMs of any fixed size, where the entries for the Pfaffian are positive integers given by simple binomial coefficient expressions. This result provides the first known case of an exact enumeration formula for an alternating sign matrix symmetry class in which a simple product formula does not seem to exist. Pfaffian formulae are also obtained for DSASM generating functions associated with several natural statistics, including the number of nonzero strictly upper triangular entries in a DSASM, the number of nonzero diagonal entries in a DSASM, and the column of the unique 1 in the first row of a DSASM. The proofs of these results involve introducing a version of the six-vertex model whose configurations are in bijection with DSASMs of fixed size, and obtaining a Pfaffian expression for its partition function. Various further results and conjectures are also obtained, including some related to the exact enumeration of off-diagonally symmetric alternating sign matrices, and some related to the asymptotic enumeration of DSASMs and other classes of alternating sign matrices.