A Central Limit Theorem for a Generalization of the Ewens Measure to Random Tuples of Commuting Permutations

Abstract We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses