| Summary: | 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 $(n^k)$.
|
|---|