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^+, we prove explicit formulas for the expected numbers of descents and inversions in the k-th powers of permutations in Sₙ for all n2k+1. We also compute the number of Grassmanian permutations in Sₙ whose k-th powers remain Grassmanian, and the number of permutations in Sₙ whose k-th powers have the maximum number of descents.