Derangement permutation matrices and orbit harmonics

Let $\mathbf{x}_{n \times n}$ be an $n \times n$ matrix of variables and let $S = \mathbb{F}[\mathbf{x}_{n \times n}]$ be the polynomial ring over these variables where $\mathbb{F}$ is a field of characteristic zero. Regard $S$ as the coordinate ring of the affine space $\mathbb{F}^{n \times n}$ of $n \times n$ $\mathbb{F}$-matrices. Let $\mathfrak{D}_n \subseteq \mathbb{F}^{n \times n}$ be the locus of derangement permutation matrices. We study the orbit harmonics quotient ring ${\bf R}(\mathfrak{D}_n) = S/\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n)$ where $\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n)$ is the associated graded ideal of the vanishing ideal $\mathbf{I}(\mathfrak{D}_n) \subseteq S$. We give an explicit generating set of $\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n),$ relate the Hilbert series of $\mathbf{R}(\mathfrak{D}_n)$ to the Foata transformation and the longest increasing subsequence statistic on $\mathfrak{S}_n$, and give an alternating sum formula for the graded $\mathfrak{S}_n$-character of $\mathbf{R}(\mathfrak{D}_n)$. Our proofs make heavy use of the mapping cone construction of homological algebra.