The Expected Number of Distinct Consecutive Patterns in a Random Permutation
Allen, Austin, Dylan Cruz Fonseca, Veronica Dobbs, Egypt Downs, Evelyn Fokuoh, Anant P. Godbole, Sebastián Papanikolaou Costa, Soto, Christopher, Lino Yoshikawa · arXiv (Cornell University) · 2020
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 in $π_n$ is $\frac{n^2}{2}(1-o(1))$. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.