Central limit theorem for descents in conjugacy classes of $S_n$

TL;DR

This paper proves a central limit theorem for the distribution of descent numbers in any conjugacy class of the symmetric group, extending previous results on specific classes and permutations.

Contribution

It generalizes existing asymptotic normality results to all conjugacy classes in the symmetric group, providing a comprehensive CLT for descent numbers.

Findings

Descent numbers in any conjugacy class are asymptotically normal.

Unified proof extending previous special cases.

Broadens understanding of permutation statistics in symmetric groups.

Abstract

The distribution of descents in fixed conjugacy classes of $S_n$ has been studied, and it is shown that its moments have interesting properties. Fulman proved that the descent numbers of permutations in conjugacy classes with large cycles are asymptotically normal, and Kim proved that the descent numbers of fixed point free involutions are also asymptotically normal. In this paper, we generalize these results to prove a central limit theorem for descent numbers of permutations in any conjugacy class of $S_n$.