Descents and inversions in powers of permutations

In this paper, we generalise several recent results by Archer and Geary on descents in powers of permutations, and confirm all their conjectures. Specifically, for all $k\in {\mathbb{Z}}^{+}$, we prove explicit formulas for the expected numbers of descents and inversions in the $k$-th powers of permutations in ${\mathcal{S}}_n$ for all $n\ge 2k+1$. We also compute the number of Grassmanian permutations in ${\mathcal{S}}_n$ whose $k$-th powers remain Grassmanian, and the number of permutations in ${\mathcal{S}}_n$ whose $k$-th powers have the maximum number of descents.