On Derangement Polynomials of Type D

Enumeration of derangements in the symmetric group Sn is classical. Ex-tensions of the enumerative results to the hyperoctahedral group Bn are combinatori-ally sound. That in the even-signed permutation group Dn remains largely unexplored. Let dDn (q) = sigma sigma is an element of DnD qmaj(sigma) be the generating function of derangements in Dn by their major indices. We study in this work properties of dDn (q), including recurrence rela-tions and factorial generating function. By proving the ratio monotonicity of dDn (q), the unimodality, log-concavity and spiral property of dDn (q) are also established.