Clustering of consecutive numbers in permutations avoiding a pattern and in separable permutations

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $σ\in S_n$ by $σ=σ_1σ_2\cdots σ_n$. 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: $\{k,k+1,\cdots, k+l-1\}=\{σ_a,σ_{a+1},\cdots, σ_{a+l-1}\}$, for some $a$. For $τ\in S_m$, let $S_n(τ)$ denote the set of $τ$-avoiding permutations in $S_n$, and let $P_n^{\text{av}(τ)}$ denote the uniform probability measure on $S_n(τ)$. 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 $P_n^{\text{av}(τ)}(A^{(n)}_{l;k})$ 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 $l\to\infty$.

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