Groupoid Cardinality and Random Permutations

TL;DR

This paper explores the connection between groupoid theory and the statistical properties of permutations, specifically focusing on the expected number of cycles, by categorifying the Cycle Length Lemma through groupoid equivalences.

Contribution

It introduces a categorification of the Cycle Length Lemma, linking permutation cycle statistics to groupoid equivalences, providing a new theoretical perspective.

Findings

Categorification of the Cycle Length Lemma via groupoid equivalences

Establishes a new theoretical framework connecting groupoids and permutation statistics

Provides insights into the structure of symmetric groups through categorical methods

Abstract

If we treat the symmetric group $S_n$ as a probability measure space where each element has measure $1/n!$, then the number of cycles in a permutation becomes a random variable. The Cycle Length Lemma describes the expected values of products of these random variables. Here we categorify the Cycle Length Lemma by showing that it follows from an equivalence between groupoids.