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Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...

Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...