4 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…

For integers $n$ and $m$, let $Pi_{n,m}$ be the locus of unordered set partitions of $[n]$ with block sizes bounded by $m$, and let $R( Pi_{n,m})$ be its orbit-harmonics graded…

Let $Pi_{(b^a)}$ and $Pi_{(a^b)}$ denote the loci of unordered set partitions whose block-size types are $(b^a)$ and $(a^b)$, respectively, and let ${\rm gr}{\bf I}( Pi)$ be the…

Let $Z\subset\mathbb{C}^n$ be any finite $S_n$-stable set such that $\mathbb{C}[x_1,\dots,x_n]/\operatorname{gr}\mathbf{I}(Z)$ is a graded $S_n$-module, where $\mathbf{I}(Z)$ is th…