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

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