Abstract
Enumeration of derangements in the symmetric group $\mathfrak{S}_n$ is classical. Extensions of the enumerative results to the hyperoctahedral group $B_n$ are combinatorially sound. That in the even-signed permutation group $D_n$ remains largely unexplored. Let $d_n^D(q) = \sum_{\sigma \in \mathcal{D}_n^D} q^{\operatorname{maj}(\sigma)}$ be the generating function of derangements in $D_n$ by their major indices. We study in this work properties of $d_n^D(q)$, including recurrence relations and factorial generating function. By proving the ratio monotonicity of $d_n^D(q)$, the unimodality, log-concavity and spiral property of $d_n^D(q)$ are also established.
Acknowledgments
The author thanks the referees for careful reading of the manuscript and valuable comments/suggestions that have led to much improved readability of this paper.
Citation
Chak-On Chow. "On Derangement Polynomials of Type $D$." Taiwanese J. Math. 27 (4) 629 - 646, August, 2023. https://doi.org/10.11650/tjm/230203
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