Descents in powers of permutations

TL;DR

This paper explores the enumeration of permutations based on the number of descents in their powers, focusing on special cases like Grassmannian permutations and permutations with maximum descents in their squares or cubes.

Contribution

It provides explicit enumeration formulas for permutations with specific descent properties in their powers, including Grassmannian permutations and those with maximum descents.

Findings

Enumerated Grassmannian permutations by descents in their square

Fully characterized permutations with one descent in their square

Counted permutations with maximum descents in their square or cube

Abstract

We consider a few special cases of the more general question: How many permutations $\pi\in\mathcal{S}_n$ have the property that $\pi^2$ has $j$ descents for some $j$? In this paper, we first enumerate Grassmannian permutations $\pi$ by the number of descents in $\pi^2$. We then consider all permutations whose square has exactly one descent, fully enumerating when the descent is "small" and providing a lower bound in the general case. Finally, we enumerate permutations whose square or cube has the maximum number of descents, and finish the paper with a few future directions for study.