Patterns in random permutations avoiding the pattern 321

TL;DR

This paper studies the distribution of pattern occurrences in random 321-avoiding permutations, revealing a non-normal limit related to Brownian excursions, and provides insights into their asymptotic behavior.

Contribution

It introduces the limiting distribution of pattern counts in 321-avoiding permutations, connecting combinatorics with stochastic processes.

Findings

Number of pattern occurrences converges to a non-normal limit distribution.

Limit can be expressed as a functional of a Brownian excursion.

Scaling by n^{m+ell} reveals the asymptotic behavior.

Abstract

We consider a random permutation drawn from the set of 321-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after scaling by $n^{m+\ell}$ where $m$ is the length of $\sigma$ and $\ell$ is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.