Let $\mathcal{P}$ be a polyomino. Two or more rooks on $\mathcal{P}$ are called non-attacking if no path of edge-adjacent cells of $\mathcal{P}$ connects any pair of them along a row or a column.
Let $F_1,F_2$ be two arrangements of $k$ non-attacking rooks.
$$ \begin{aligned} F_1 \sim F_2 \quad &\text{if } F_2 \text{ can be obtained from } F_1 \text{ by moving two or more rooks from}\\ &\text{diagonal to anti-diagonal positions, or vice versa, within $\mathcal{P}$.} \end{aligned} $$
Let $\tilde{r}(k,\mathcal{P})$ denote the number of equivalence classes of the arrangements of $k$ non-attacking rooks. The maximum number of non-attacking rooks that can be placed on a polyomino $\mathcal{P}$ is called rook number of $\mathcal{P}$, denoted by $r(\mathcal{P})$.
We can define the following polynomial:
$$
\tilde{r}_{\mathcal{P}}(t) = \sum_{k=0}^{r(\mathcal{P})} \tilde{r}(k,\mathcal{P})t^k
$$
The figure below illustrates four $3$-rook configurations in $\mathcal{P}$ that are equivalent under switches.
The rook number of $\mathcal{P}$ is four, and $ \tilde{r}_{\mathcal{P}}(t) = 1 + 12t + 30t^2 + 16t^3 + t^4. $
Question. It is well known that the classical rook polynomial is related to graph theory, as it coincides with the matching polynomial of a bipartite graph. My question is whether the variant of the rook polynomial described above is also related to graph theory.
Motivation behind the question. I encountered this variant of the rook polynomial in a context related to Combinatorial Commutative Algebra, where it provides a combinatorial interpretation of the Hilbert series of certain binomial ideals associated with polyominoes, namely the determinantal ideals generated by inner $2$-minors of matrices that can be represented as polyominoes. You can see
Jahangir, Rizwan; Navarra, Francesco, Shellable simplicial complex and switching rook polynomial of frame polyominoes, J. Pure Appl. Algebra 228, No. 6, Article ID 107576, 25 p. (2024). ZBL1533.05053.
and the references there in. Given this, I also wondered whether this version of the rook polynomial might admit connections with graph theory, in analogy with the classical rook polynomial.
