Descents in powers of permutations

Abstract

We consider a few special cases of the more general question: How many permutations $\ddot{I}€\in {\mathcal{S}}_n$ have the property that $\ddot{I}{€}^2$ has $j$ descents for some $j$? In this paper, we first enumerate Grassmannian permutations $\ddot{I}€$ by the number of descents in $\ddot{I}{€}^2$. 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.