Groupoid Cardinality and Random Permutations
John C. Baez
TL;DR
The paper develops a categorified perspective on the Cycle Length Lemma for random permutations by leveraging finite groupoid cardinality. It proves a key equivalence $\mathsf{C}_{\vec{p}} \simeq \mathsf{Perm}_{n-|\vec{p}|} \times \prod_{k=1}^n \mathsf{B}(\mathbb{Z}/k)^{p_k}$, linking the counting of disjoint cycles to classifying spaces $\mathsf{B}(\mathbb{Z}/k)$ and to the action groupoid $\Perm_n$. From this equivalence, the lemma follows as a cardinality identity, with $|\mathsf{B}(\mathbb{Z}/k)| = 1/k$ and $|\Perm_n|=1$, and the framework clarifies how $E\left(\prod_{k} c_k^{\underline{p_k}}\right) = \prod_{k} 1/k^{p_k}$ for $|\vec{p}|\le n$ and vanishes otherwise. The work also generalizes to arbitrary finite groups via the category of elements, showing how conjugation-equivariant structures on group elements can be analyzed categorically, with potential extensions to groups like $\mathrm{GL}(n,\mathbb{F}_q)$. Overall, the paper illuminates a deep connection between permutation statistics and groupoid theory, enabling new avenues for proving and generalizing cycle-type results.
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.