A symmetric unimodal decomposition of the derangement polynomial of type $B$

Abstract

The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n(x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of $d_n (x)$ as the local $h$-polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition and a new formula for the Eulerian polynomial of type $B$. The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness.