Distributions of descent and flag major index on colored permutation conjugacy classes

Determine the distributions of the descent statistic \(\des_{n,r}\) and the flag major index statistic \(\fmaj_{n,r}\) on conjugacy classes of the colored permutation group \(\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr\mathfrak{S}_n\).

Background

The paper establishes that, for conjugacy classes of $\mathfrak{S}_{n,r}$ with sufficiently long cycles, every fixed moment of $\des_{n,r}$ and $\fmaj_{n,r}$ agrees with the corresponding moment on the whole group. It also derives asymptotic normality for sequences of conjugacy classes whose numbers of cycles of each fixed length tend to zero.

The exact probability distributions on conjugacy classes remain to be determined. The authors note that analogous distributional results are known for the ordinary descent statistic on conjugacy classes of $\mathfrak{S}_n$ and for a different descent statistic on the hyperoctahedral group $B_n$, but those results do not directly resolve the colored statistics considered here.