Let L n denote the set of all paths from [0, 0] to [n, n] which consist of either unit north steps N or unit east steps E or, equivalently, the set of all words L ∈ {E, N } * with n E's and n N 's. Given L ∈ L n and a subset A of [n] = {1, . . . , n}, we let ps L (A) denote the word that results from L by removing the i th occurrence of E and the i th occurrence of N in L for all i ∈ [n] − A, reading from left to right. Then we say that a paired pattern P ∈ L k occurs in L if there is some A ⊆ [n] of size k such that ps L (A) = P . In this paper, we study the generating functions of paired pattern matching in L n .
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 suggestions
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