Restricted inversion polynomials

For a finite subset $I$ of positive integers, the descent polynomial $\mathcal{D}(I;n)$ counts the number of permutations in $S_n$ that have descent set $I$. We generalize descent polynomials by considering permutations with a specific subset $S$ of common inversions called $\mathbf{h}$-inversions, where $\mathbf{h}=(\mathbf{h}(1),\mathbf{h}(2),\dots )$ is a weakly increasing sequence of positive integers such that $\mathbf{h}(i)>i$. We prove that this more general count, denoted by ${\mathcal{I}}_{\mathbf{h}}(S;n)$, is also a polynomial. We give three explicit expansions for ${\mathcal{I}}_{\mathbf{h}}(S;n)$, prove the coefficients for two of these expansions are log-concave, and define a graded generalization.