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

Let denote the set of permutations of [n] for which the set of l consecutive numbers appears in a set of consecutive positions. Under the uniform probability measure Pn on Sn, one has.In one part of this paper we consider the probability of clustering of consecutive numbers under Mallows distributions Because of a duality, it suffices to consider We show that for is of order uniformly over all sequences Thus, letting denote the number of sets of I consecutive numbers appearing in sets of consecutive positions, we have lira We also consider the cases. 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 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.