Pan, Ran and Remmel, Jeffrey B. - Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

dmtcs:1490 - Discrete Mathematics & Theoretical Computer Science, May 20, 2016, Vol. 18 no. 2, Permutation Patterns 2015
Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

Authors: Pan, Ran and Remmel, Jeffrey B.

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 $(n^k)$.


Source : oai:arXiv.org:1510.08190
Volume: Vol. 18 no. 2, Permutation Patterns 2015
Section: Permutation Patterns
Published on: May 20, 2016
Submitted on: May 18, 2016
Keywords: Mathematics - Combinatorics


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