3 problems

Archer–Geary conjecture. For all but the first few values of $n$, the expected numbers of descents in $\pi^2$ and in $\pi^3$ are both

For $n\geq 3$, let $\pi$ range uniformly over the symmetric group ${\mathcal S}_n$, and let $\operatorname{des}(\pi^2)$ denote the number of descents of the square of $\pi$. The ex…

The generating-function conjecture. The ordinary generating function is