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

Ross G. Pinsky

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 $l\ge 2$ an integer, let $A^{(n)}_{l;k}\subset S…