On Derangement Polynomials of Type $D$

Enumeration of derangements in the symmetric group $\mathfrak{S}_n$ is classical. Extensions enumerative results to hyperoctahedral $B_n$ are combinatorially sound. That even-signed permutation $D_n$ remains largely unexplored. Let $d_n^D(q) = \sum_{\sigma \in \mathcal{D}_n^D} q^{\operatorname{maj}(\sigma)}$ be generating function by their major indices. We study this work properties $d_n^D(q)$, including recurrence relations and factorial function. By proving ratio monotonicity unimodality, log-concavity spiral property $d_n^D(q)$ also established.