Key-avoidance for alternating sign matrices

TL;DR

This paper systematically studies key-avoidance patterns in alternating sign matrices (ASMs), linking pattern-avoidance to permutation classes, Catalan numbers, and Schubert polynomials, and extends known combinatorial results.

Contribution

It introduces the concept of key-avoidance in ASMs, enumerates ASMs avoiding specific permutation patterns, and connects these to known combinatorial structures and identities.

Findings

ASMs with key avoiding 231 are permutations

ASMs avoiding 312 and 321 are counted by Catalan numbers

ASMs avoiding 312 relate to gapless monotone triangles

Abstract

We initiate a systematic study of key-avoidance on alternating sign matrices (ASMs) defined via pattern-avoidance on an associated permutation called the \emph{key} of an ASM. We enumerate alternating sign matrices whose key avoids a given set of permutation patterns in several instances. We show that ASMs whose key avoids $231$ are permutations, thus any known enumeration for a set of permutation patterns including $231$ extends to ASMs. We furthermore enumerate by the Catalan numbers ASMs whose key avoids both $312$ and $321$. We also show ASMs whose key avoids $312$ are in bijection with the gapless monotone triangles of [Ayyer, Cori, Gouyou-Beauchamps 2011]. Thus key-avoidance generalizes the notion of $312$-avoidance studied there. Finally, we enumerate ASMs with a given key avoiding $312$ and $321$ using a connection to Schubert polynomials, thereby deriving an interesting Catalan identity.