117 research outputs found

### 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

### Quadrant marked mesh patterns in 123-avoiding permutations

Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric
group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked
mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$
points to the right of $\sigma_i$ in $\sigma$ which are greater than
$\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are
greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in
$\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the
right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev,
Remmel, and Tiefenbruck systematically studied the distribution of the number
of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The
operation of reverse and complement on permutations allow one to translate
their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$
matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this
paper, we study the distribution of the number of matches of
$\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit
recurrence relations to enumerate our objects which can be used to give closed
forms for the generating functions associated with such distributions. In many
cases, we provide combinatorial explanations of the coefficients that appear in
our generating functions

### Descent c-Wilf Equivalence

Let $S_n$ denote the symmetric group. For any $\sigma \in S_n$, we let
$\mathrm{des}(\sigma)$ denote the number of descents of $\sigma$,
$\mathrm{inv}(\sigma)$ denote the number of inversions of $\sigma$, and
$\mathrm{LRmin}(\sigma)$ denote the number of left-to-right minima of $\sigma$.
For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on
permutations, we say two permutations $\alpha$ and $\beta$ in $S_j$ are
$(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating
function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which
have no consecutive occurrences of $\alpha$ equals the generating function of
$\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no
consecutive occurrences of $\beta$. We give many examples of pairs of
permutations $\alpha$ and $\beta$ in $S_j$ which are $\mathrm{des}$-c-Wilf
equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and
$(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we
will show that if $\alpha$ and $\beta$ are minimally overlapping permutations
in $S_j$ which start with 1 and end with the same element and
$\mathrm{des}(\alpha) = \mathrm{des}(\beta)$ and $\mathrm{inv}(\alpha) =
\mathrm{inv}(\beta)$, then $\alpha$ and $\beta$ are
$(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431

- β¦