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

arXiv2109.09370MaRDI QIDQ6378052

Ross G. Pinsky

Publication date: 20 September 2021

Abstract: Let $S_n$ denote the set of permutations of $[n]:=1,cdots,n$, and denote a permutation $sigmain{S}_n$ by $sigma=sigm{a}_{1}sigm{a}_{2}cdotssigm{a}_n$. For $lge2$ an integer, let $A_{l;k}^{(n)}subset{S}_n$ denote the event that the set of $l$ consecutive numbers $k,k+1,cdots,k+l-1$ appears in a set of consecutive positions: $k,k+1,cdots,k+l-1=sigm{a}_a,sigm{a}_{a+1},cdots,sigm{a}_{a+l-1}$, for some $a$. For $auin{S}_m$, let $S_n(au)$ denote the set of $au$-avoiding permutations in $S_n$, and let $P_n^{extav(au)}$ denote the uniform probability measure on $S_n(au)$. Also, let $S_n^{extsep}$ denote the set of separable permutations in $S_n$, and let $P_n^{extsep}$ denote the uniform probability measure on $S_n^{extsep}$. We investigate the quantities $P_n^{extav(au)}(A_{l;k}^{(n)})$ and $P_n^{extsep}(A_{l;k}^{(n)})$ for fixed $n$, and the limiting behavior as $noinfty$. We also consider the asymptotic properties of this limiting behavior as $loinfty$.

Mathematics Subject Classification ID

Permutations, words, matrices (05A05) Combinatorial probability (60C05)

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