Asymptotic Behaviour of the Containment of Certain Mesh Patterns

TL;DR

This paper investigates the asymptotic behavior of permutations containing specific mesh patterns, providing exact enumeration and demonstrating the wide range of their occurrence proportions as permutation length increases.

Contribution

It offers new general results on the asymptotic containment of mesh patterns and detailed analysis for patterns of length four, expanding understanding of their distribution.

Findings

Proportion of permutations containing certain mesh patterns varies widely between 0 and 1.

Exact enumeration results are provided for some mesh patterns.

General asymptotic results apply to mesh patterns of any length.

Abstract

We present some results on the proportion of permutations of length $n$ containing certain mesh patterns as $n$ grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between $0$ and $1$.