Distributions of mesh patterns of short lengths on king permutations

TL;DR

This paper systematically studies the distribution of 22 short-length mesh patterns within king permutations, building on prior work on mesh patterns and king permutation enumeration.

Contribution

It introduces the first systematic analysis of mesh pattern distributions specifically on king permutations, focusing on 22 short-length patterns.

Findings

Distribution formulas for 22 mesh patterns on king permutations

Extension of mesh pattern analysis to a new permutation class

Connections to existing enumeration results for king permutations

Abstract

Br\"{a}nd\'{e}n and Claesson introduced the concept of mesh patterns in 2011, and since then, these patterns have attracted significant attention in the literature. Subsequently, in 2015, Hilmarsson \emph{et al.} initiated the first systematic study of avoidance of mesh patterns, while Kitaev and Zhang conducted the first systematic study of the distribution of mesh patterns in 2019. A permutation $\sigma = \sigma_1 \sigma_2 \cdots \sigma_n$ in the symmetric group $S_n$ is called a king permutation if $\left| \sigma_{i+1}-\sigma_i \right| > 1$ for each $1 \leq i \leq n-1$. Riordan derived a recurrence relation for the number of such permutations in 1965. The generating function for king permutations was obtained by Flajolet and Sedgewick in 2009. In this paper, we initiate a systematic study of the distribution of mesh patterns on king permutations by finding distributions for 22 mesh patterns of short length.