Let $S_n$ denote the set of permutations of $[n]:=\{ 1,\cdots, n\}$, and denote a permutation $\sigma\in S_n$ by $\sigma=\sigma_1\sigma_2\cdots \sigma_n$. For $l\ge2$ an integer, let $A^{ (n)}_{ l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{ k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions: $\{ k,k+1,\cdots, k+l-1\}=\{ \sigma_a,\sigma_{ a+1},\cdots, \sigma_{ a+l-1}\}$, for some $a$. For $\tau\in S_m$, let $S_n(\tau)$ denote the set of $\tau$-avoiding permutations in $S_n$, and let $P_n^{ \text{ av}(\tau)}$ denote the uniform probability measure on $S_n(\tau)$. Also, let $S_n^{ \text{ sep}}$ denote the set of separable permutations in $S_n$, and let $P_n^{ \text{ sep}}$ denote the uniform probability measure on $S_n^{ \text{ sep}}$. We investigate the quantities $P_n^{ \text{ av}(\tau)}(A^{ (n)}_{ l;k})$ and $P_n^{ \text{ sep}}(A^{ (n)}_{ l;k})$ for fixed $n$, and the limiting behavior as $n\to\infty$. We also consider the asymptotic properties of this limiting behavior as $l\to\infty$.
Categories: math.CO math.PR