Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns

TL;DR

This paper derives explicit and asymptotic formulas for the expected number of consecutive pattern occurrences in permutations avoiding certain patterns of length three or more, including simple and separable permutations.

Contribution

It provides the first explicit formulas for these expectations across all pattern-avoiding classes of permutations, extending to asymptotic behaviors as permutation size grows.

Findings

Explicit formulas for expected pattern occurrences in pattern-avoiding permutations.

Asymptotic formulas for large permutation sizes and pattern lengths.

Results include special classes like separable permutations.

Abstract

For $\eta\in S_3$, let $S_n^{\text{av}(\eta)}$ denote the set of permutations in $S_n$ that avoid the pattern $\eta$, and let $E_n^{\text{av}(\eta)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\text{av}(\eta)}$. For $n\ge k\ge2$ and $\tau\in S_k^{\text{av}(\eta)}$, let $N_n^{(k)}(\sigma)$ denote the number of occurrences of $k$ consecutive numbers appearing in $k$ consecutive positions in $\sigma\in S_n^{\text{av}(\eta)}$, and let $N_n^{(k;\tau)}(\sigma)$ denote the number of such occurrences for which the order of the appearance of the $k$ numbers is the pattern $\tau$. We obtain explicit formulas for $E_n^{\text{av}(\eta)}N_n^{(k;\tau)}$ and $E_n^{\text{av}(\eta)}N_n^{(k)}$, for all $2\le k\le n$, all $\eta\in S_3$ and all $\tau\in S_k^{\text{av}(\eta)}$. These exact formulas then yield asymptotic formulas as $n\to\infty$ with $k$ fixed, and as $n\to\infty$ with $k=k_n\to\infty$. We also obtain analogous results for $S_n^{\text{av}(\eta_1,\cdots,\eta_r)}$, the subset of $S_n$ consisting of permutations avoiding the patterns $\{\tau_i\}_{i=1}^r$, where $\tau_i\in S_{m_i}$, in the case that $\{\tau_i\}_{i=1}^n$ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to $r=2$, $\tau_1=2413,\tau_2=3142$.