Groupoid Cardinality and Random Permutations

2024/12/20 by Baez, John C. · 1 citation
#Category Theory (math.CT) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.16386

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.

Cited by

• Free loop spaces and the Cauchy--Frobenius Lemma

Related

• The Cycle Structure of Permutations Without Long Cycles
• Representing Random Permutations as the Product of Two Involutions
• Local probabilities for random permutations without long cycles
• The cycle structure of compositions of random involutions
• Ewens sampling and invariable generation