Moments of Colored Permutation Statistics on Conjugacy Classes

In this paper, we consider the moments of statistics on conjugacy classes of the colored permutation groups $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We first show that any fixed moment coincides on all conjugacy classes where all cycles have sufficiently long length. Additionally, for permutation statistics that can be realized via a process we call symmetric extensions, these moments are polynomials in $n$. Finally, for the descent statistic on the hyperoctahedral group $B_n\cong \mathfrak{S}_{n,2}$, we show that its distribution on conjugacy classes without short cycles satisfies a central limit theorem. Our results build on and generalize previous work of Fulman (\textit{J. Comb. Theory Ser. A.}, 1998), Hamaker and Rhoades (arXiv, 2022), and Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin (arXiv, 2023). In particular, our techniques utilize the combinatorial framework introduced by Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin.