Colored Permutation Statistics by Conjugacy Class

Jesse Campion Loth, Michael Levet, Kevin Liu, Sheila Sundaram, Mei Yin

Published 2023-05-19Version 1

In this paper, we consider the moments of permutation statistics on conjugacy classes of colored permutation groups. We first show that when the cycle lengths are sufficiently large, the moments of arbitrary permutation statistics are independent of the conjugacy class. For permutation statistics that can be realized via $\textit{symmetric}$ constraints, we show that for a fixed number of colors, each moment is a polynomial in the degree $n$ of the $r$-colored permutation group $\mathfrak{S}_{n,r}$. Hamaker & Rhoades (arXiv 2022) established analogous results for the symmetric group as part of their far-reaching representation-theoretic framework. Independently, Campion Loth, Levet, Liu, Stucky, Sundaram, & Yin (arXiv, 2023) arrived at independence and polynomiality results for the symmetric group using instead an elementary combinatorial framework. Our techniques in this paper build on this latter elementary approach.