The Expected Number of Distinct Consecutive Patterns in a Random Permutation

Abstract

Let ${\pi }_n$ 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 in ${\pi }_n$ is $\frac{n^2}{2}(1-o(1))$. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.

12 pages, 2 figures