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arXiv · 2406.07311

Pattern containment in random permutations

Jonna Gill
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Abstract

This paper studies permutation statistics that count occurrences of patterns. Their expected values on a product of $t$ permutations chosen randomly from $Γ\subseteq S_{n}$, where $Γ$ is a union of conjugacy classes, are considered. Hultman has described a method for computing such an expected value, denoted $\mathbb{E}_Γ(s,t)$, of a statistic $s$, when $Γ$ 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 $\mathbb{E}_Γ(s,t)$ for all the above statistics, and for all functions on $S_{n}$ that are linear combinations of them.

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Jonna Gill. 2024-06-11. Pattern containment in random permutations. https://arxiv.org/abs/2406.07311

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