Pattern containment in random permutations

TL;DR

This paper develops a method to compute expected pattern occurrences in random permutations, focusing on expressing pattern-counting statistics as linear combinations of irreducible characters, enabling broader calculations of permutation statistics.

Contribution

It introduces a procedure to express means of pattern-counting statistics as linear combinations of irreducible characters of symmetric groups, facilitating expected value computations.

Findings

Developed a method for calculating linear combinations for pattern statistics

Computed expressions for classical and vincular patterns of length 3

Enabled calculation of expected pattern counts in random permutations

Abstract

This paper studies permutation statistics that count occurrences of patterns. Their expected values on a product of $t$ permutations chosen randomly from $\Gamma \subseteq S_{n}$, where $\Gamma$ is a union of conjugacy classes, are considered. Hultman has described a method for computing such an expected value, denoted $\mathbb{E}_{\Gamma}(s,t)$, of a statistic $s$, when $\Gamma$ is a union of conjugacy classes of $S_{n}$. The only prerequisite is that the mean of $s$ over the conjugacy classes is written as a linear combination of irreducible characters of $S_{n}$. Therefore, the main focus of this article is to express the means of pattern-counting statistics as such linear combinations. A procedure for calculating such expressions for statistics counting occurrences of classical and vincular patterns of length 3 is developed, and is then used to calculate all these expressions. The results can be used to compute $\mathbb{E}_{\Gamma}(s,t)$ for all the above statistics, and for all functions on $S_{n}$ that are linear combinations of them.