The expected number of distinct consecutive patterns in a random permutation

Abstract

Abstract Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2,…, n} in π n is n 2 2 ( 1 - o ( 1 ) ) {{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right) as n → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.