A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

TL;DR

This paper establishes a central limit theorem for the number of joint orbits in random tuples of commuting permutations, extending classical results and introducing a generalized Ewens measure for such permutations.

Contribution

It generalizes the CLT for cycle counts to tuples of commuting permutations under a new weighted measure, using advanced saddle point analysis techniques.

Findings

Proves a CLT for joint orbits of commuting permutations

Extends Goncharov's classic CLT to a broader setting

Introduces a generalized Ewens measure for permutation tuples

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 saddle point analysis, in a context related to the Hardy-Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.