Descents in powers of permutations

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.