Groupoid Cardinality and Random Permutations

If we treat the symmetric group $S_n$ as a probability measure space where each element has measure $1\mathrm{/}n!$, 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.