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

Abstract

Let $A_{l;k}^{(n)}\subset S_n$ 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 uniform probability measure $P_n$ on $S_n$, one has $P_n(A_{l;k}^{(n)})\sim \frac{l!}{n^{l-1}}$ as $n\to \mathrm{\infty }$. 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 $q_n=1-\frac{c}{n^{\mathit{\alpha }}}$, with $c>0$ and $\mathit{\alpha }\in (0,1)$, $P_n^q(A_{l;k_n}^{(n)})$ is of order $\frac{1}{n^{\mathit{\alpha }(l-1)}}$, uniformly over all sequences ${\{k_n\}}_{n=1}^{\mathrm{\infty }}$. Thus, letting $N_l^{(n)}={\sum }_{k=1}^{n-l+1}{1}_{A_{l;k}^{(n)}}$ denote the number of sets of l consecutive numbers appearing in sets of consecutive positions, we have

$$\underset{n\to \mathrm{\infty }}{lim}{E}_n^{q_n}{N}_l^{(n)}=\left\{\begin{array}{l}\mathrm{\infty },\text{if}l<\frac{1+\mathit{\alpha }}{\mathit{\alpha }};\\ 0,\text{if}l>\frac{1+\mathit{\alpha }}{\mathit{\alpha }}.\end{array}\right..$$

We also consider the cases $\mathit{\alpha }=1$ and $\mathit{\alpha }>1$. In the other part of the paper we consider general p-shifted distributions, $P_n^{{\{p_j\}}_{j=1}^{\mathrm{\infty }}}$, of which the Mallows distribution is a particular case. We calculate explicitly the quantity

$$\underset{l\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{P}_n^{{\{p_j\}}_{j=1}^{\mathrm{\infty }}}(A_{l;k_n}^{(n)})=\underset{l\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{P}_n^{{\{p_j\}}_{j=1}^{\mathrm{\infty }}}(A_{l;k_n}^{(n)})$$

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 fixed parameter $q\ne 1$.