2 problems

For $0\le r\le\min\{m,n\}$, let ${\mathcal{Z}}_{n,m,r}$ be the locus of $n\times m$ rook placements with exactly $r$ rooks, and let $R({\mathcal{Z}}_{n,m,r})$ denote its orbit harm…

Let $\lambda\in\mathcal{D}_n$ be a fixed partition, and let $\varepsilon>0$. For a uniformly random rook placement $\pi\sim\operatorname{Unif}(\operatorname{RP}(N\odot\lambda))$, w…