Enumeration of pattern-avoiding alternating sign matrices: An asymptotic dichotomy

TL;DR

This paper classifies the asymptotic growth of alternating sign matrices avoiding specific permutation patterns, revealing exponential and super-exponential bounds, and enumerates matrices with a fixed number of negative ones.

Contribution

It provides a complete asymptotic classification for pattern-avoiding alternating sign matrices and introduces new bounds and exact counts for matrices with constrained negative entries.

Findings

Exponential upper bounds for avoiding one of eleven patterns.

Super-exponential lower bounds for other pattern avoidance classes.

Exact enumeration of matrices with three negative ones avoiding certain patterns.

Abstract

We completely classify the asymptotic behavior of the number of alternating sign matrices classically avoiding a single permutation pattern, in the sense of [Johansson and Linusson 2007]. In particular, we give a uniform proof of an exponential upper bound for the number of alternating sign matrices classically avoiding one of eleven particular patterns, and a super-exponential lower bound for all other single-pattern avoidance classes. We also show that for any fixed integer $k$, there is an exponential upper bound for the number of alternating sign matrices that classically avoid any single permutation pattern and contain precisely $k$ negative ones. Finally, we prove that there must be at most $3$ negative ones in an alternating sign matrix which classically avoids both $2143$ and $3412$, and we exactly enumerate the number of them with precisely $3$ negative ones.