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

Ran Pan,Jeffrey B Remmel

doi:10.46298/dmtcs.1315

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 $(n^k)$.

Highlights

Let Sn denote the set of permutations of [n] = {1, . . . , n}
One way to think of elements in C⊆k[n−1] is as column strict arrays which were studied by Harmse and Remmel in [6] and [7]
A column-strict array P is a filling of a k × n rectangular array with 1, 2, · · ·, kn such that elements increase from bottom to top in each column

Summary

Introduction

If F is any filling of a k × n-rectangle with distinct positive integers such that elements in each column increase, reading from bottom to top, we let red(F ) denote the element of Fn,k which results from F by replacing the ith smallest element of F by i. By (3), L1 ≈ 0.282, L2 ≈ 0.822, L3 ≈ 0.950, L4 ≈ 0.986, L5 ≈ 0.996, · · · It follows from our observations in the previous section that a pattern P ∈ Fn,k is overlapping if and only if there exists a unique integer i such that P is overlapping at position i and the reduction of the first i columns of P is minimally overlapping with 2. This is not a coincidence because (3) is almost the same as (6) and (7) and we know an,k is close to 1 when n is large

Standard tableaux of rectangular shapes

Generalized Euler permutations

Findings

Open questions