The Expected Number of Distinct Consecutive Patterns in a Random Permutation

TL;DR

This paper analyzes the expected count of distinct consecutive patterns in a random permutation, revealing that such permutations nearly perfectly pack these patterns as the size grows.

Contribution

It provides an asymptotic estimate for the expected number of distinct consecutive patterns in a random permutation, highlighting near-perfect packing behavior.

Findings

Expected number of distinct consecutive patterns is approximately n^2/2

Random permutations pack consecutive patterns near-perfectly

Asymptotic behavior confirmed for large n

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.