Asymptotic descent distribution on arbitrary colored conjugacy classes

Determine the asymptotic distribution of the descent statistic \(\des_{n,r}\) on arbitrary conjugacy classes of the colored permutation group \(\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr\mathfrak{S}_n\).

Background

The paper proves asymptotic normality of $\des_{n,r}$ only for sequences of conjugacy classes in which the number of cycles of every fixed length tends to zero. Thus, the result applies to conjugacy classes with sufficiently long cycles, rather than to arbitrary conjugacy classes.

The authors point to the known result that the ordinary descent statistic on arbitrary conjugacy classes of Sn\mathfrak{S}_n is asymptotically normal and ask for the corresponding characterization in the colored setting. Resolving this problem would extend the paper’s asymptotic results beyond the long-cycle regime.

References

Hence, one can consider the corresponding problem for $\des_{n,r}$ on arbitrary conjugacy classes of $\mathfrak{S}_{n,r}$, and results from \cref{prob:1} may be useful for this.

\begin{problem}\label{prob:2} Determine the asymptotic distribution for $\des_{n,r}$ on arbitrary conjugacy classes of $\mathfrak{S}_{n,r}$. \end{problem}

— Descents and flag major index on conjugacy classes of colored permutation groups without short cycles  (2503.02990 - Liu et al., 4 Mar 2025) in Section 6, Conclusion, Problem 2