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.
Keywords: groupoid, random permutation, cycle
2020 MSC: 05A05, 05A19, 20L05
Theory and Applications of Categories, Vol. 44, 2025, No. 14, pp 410-419.
Published 2025-05-16.
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