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arXiv · 2411.07662

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

Mathilde Bouvel·Eric S. Egge·Rebecca N. Smith·Jessica Striker·Justin M. Troyka
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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.

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Mathilde Bouvel, Eric S. Egge, Rebecca N. Smith, Jessica Striker, Justin M. Troyka. 2024-11-12. Enumeration of pattern-avoiding alternating sign matrices: An asymptotic dichotomy. https://arxiv.org/abs/2411.07662

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