Clustering of consecutive numbers in permutations avoiding a pattern and in separable permutations

TL;DR

This paper studies the probability of consecutive number clusters in permutations avoiding certain patterns and in separable permutations, analyzing their behavior for large permutation sizes and cluster lengths.

Contribution

It provides new insights into the asymptotic probabilities of consecutive number clusters in pattern-avoiding and separable permutations, including their limiting behaviors.

Findings

Asymptotic probabilities are derived for pattern-avoiding permutations.

Limiting behavior of cluster probabilities as permutation size grows.

Analysis of cluster size growth as permutations become large.

Abstract

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $\sigma\in S_n$ by $\sigma=\sigma_1\sigma_2\cdots \sigma_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions: $\{k,k+1,\cdots, k+l-1\}=\{\sigma_a,\sigma_{a+1},\cdots, \sigma_{a+l-1}\}$, for some $a$. For $\tau\in S_m$, let $S_n(\tau)$ denote the set of $\tau$-avoiding permutations in $S_n$, and let $P_n^{\text{av}(\tau)}$ denote the uniform probability measure on $S_n(\tau)$. Also, let $S_n^{\text{sep}}$ denote the set of separable permutations in $S_n$, and let $P_n^{\text{sep}}$ denote the uniform probability measure on $S_n^{\text{sep}}$. We investigate the quantities $P_n^{\text{av}(\tau)}(A^{(n)}_{l;k})$ and $P_n^{\text{sep}}(A^{(n)}_{l;k})$ for fixed $n$, and the limiting behavior as $n\to\infty$. We also consider the asymptotic properties of this limiting behavior as $l\to\infty$.