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

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 $(n^k)$.Comment: Accepted by Discrete Math and Theoretical Computer Science. Thank referees' for their suggestion

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

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 $(n^k)$

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

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping ifany two consecutive occurrences of $\tau$ in a permutation $\sigma$ can shareat most one element. B\'ona \cite{B} showed that the proportion of minimaloverlapping 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 studythe class of permutations $\sigma \in S_{kn}$ whose descent set is contained inthe 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 ofthe minimal overlapping permutations for such classes of permutations and westudy the proportion of minimal overlapping patterns for each such class. Weshow that the proportion of minimal overlapping permutations in such classesapproaches $1$ as $k$ goes to infinity. We also study the proportion of minimaloverlapping patterns in standard Young tableaux of shape $(n^k)$.Comment: Accepted by Discrete Math and Theoretical Computer Science. Thank referees' for their suggestion