Descents and inversions in powers of permutations

TL;DR

This paper extends recent research on descents and inversions in permutation powers, providing explicit formulas and confirming conjectures for various permutation classes and their properties.

Contribution

It generalizes previous results by Archer and Geary, offering explicit formulas and confirming conjectures about descents and inversions in powers of permutations.

Findings

Explicit formulas for expected descents and inversions in permutation powers

Number of Grassmanian permutations with Grassmanian powers

Permutations with maximum descents in their powers

Abstract

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 $n\geq2k+1$. 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.