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arXiv · 2411.18131

Distributions of mesh patterns of short lengths on king permutations

Dan Li·Philip B. Zhang
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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.

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Dan Li, Philip B. Zhang. 2024-11-27. Distributions of mesh patterns of short lengths on king permutations. https://arxiv.org/abs/2411.18131

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