Clustering of consecutive numbers in permutations under Mallows distributions and super-clustering under general $p$-shifted distributions

Abstract

Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on $S_n$, one has $P_n(A^{(n)}_{l;k})\sim\frac{l!}{n^{l-1}}$ as $n\to\infty$. In one part of this paper we consider the probability of clustering of consecutive numbers under Mallows distributions $P_n^q$, $q>0$. Because of a duality, it suffices to consider $q\in(0,1)$. We show that for $q_n=1-\frac c{n^α}$, with $c>0$ and $α\in(0,1)$, $P_n^q(A^{(n)}_{l;k_n})$ is on the order $\frac1{n^{α(l-1)}}$, uniformly over all sequences $\{k_n\}_{n=1}^\infty$. Thus, letting $N^{(n)}_l=\sum_{k=1}^{n-l+1}1_{A^{(n)}_{l;k}}$ denote the number of sets of $l$ consecutive numbers appearing in sets of consecutive positions, we have \begin{equation*} \lim_{n\to\infty} E_n^{q_n}N^{(n)}_l = \begin{cases}\infty,\ \text{if}\ l<\frac{1+α}α;\\ 0,\ \text{if} \ l>\frac{1+α}α. \end{cases}. \end{equation*} We also consider the cases $α=1$ and $α>1$. In the other part of the paper we consider general $p$-shifted distributions, of which the Mallows distribution is a particular case. We calculate explicitly the quantity $\lim_{l\to\infty} \liminf_{n\to\infty}P_n^q(A^{(n)}_{l;k_n}) = \lim_{l\to\infty}\limsup_{n\to\infty}P_n^q(A^{(n)}_{l;k_n})$ in terms of the $p$-distribution. When this quantity is positive, we say that super-clustering occurs. In particular, super-clustering occurs for the Mallows distribution with parameter $q\neq1$. We also give a new characterization of $p$-shifted distributions.