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

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