Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

TL;DR

This paper investigates the asymptotic behavior of minimal overlapping patterns in various combinatorial structures, showing that their proportion approaches 1 in certain classes as parameters grow large.

Contribution

It introduces and analyzes the concept of minimal overlapping patterns in generalized Euler permutations, standard tableaux, and column strict arrays, revealing their asymptotic properties.

Findings

Proportion of minimal overlapping permutations approaches 1 as k increases.

Provides bounds and asymptotic results for minimal overlapping patterns in different classes.

Extends the concept of minimal overlapping patterns to new combinatorial objects.

Abstract

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $\sigma$, we let $\text{Des}(\sigma)$ denote the set of descents of $\sigma$. We study the class of permutations $\sigma \in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $\sigma$ such that $\text{Des}(\sigma) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$.