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

Abstract

Let Al;k(n)⊂Sn denote the set of permutations of [n] for which the set of l consecutive numbers {k,k+1,⋯,k+l−1} appears in a set of consecutive positions. Under the uniform probability measure Pn on Sn, one has Pn(Al;k(n))∼l! nl−1 as n→∞. In one part of this paper we consider the probability of clustering of consecutive numbers under Mallows distributions Pnq, q>0. Because of a duality, it suffices to consider q∈(0,1). We show that for qn=1−c nα, with c>0 and α∈(0,1), Pnq(Al;kn(n)) is of order 1 nα(l−1), uniformly over all sequences {kn}n=1∞. Thus, letting Nl(n)= ∑k=1n−l+11Al;k(n) denote the number of sets of l consecutive numbers appearing in sets of consecutive positions, we have limn→∞EnqnNl(n)= ∞,ifl<1+α α;0,ifl>1+α α.. We also consider the cases α=1 and α>1. In the other part of the paper we consider general p-shifted distributions, Pn{pj}j=1∞, of which the Mallows distribution is a particular case. We calculate explicitly the quantity liml→∞lim infn→∞Pn{pj}j=1∞(Al;kn(n))=lim l→∞lim supn→∞Pn{pj}j=1∞(Al;kn(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≠1.