Descents in powers of permutations

We consider a few special cases of the more general question: How many permutations $\pi \in S_n$ have the property that ${\pi }^2$ has j descents for some j? In this paper, we first enumerate Grassmannian permutations π 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.