23 research outputs found
Odd length for even hyperoctahedral groups and signed generating functions
We define a new statistic on the even hyperoctahedral groups which is a
natural analogue of the odd length statistic recently defined and studied on
Coxeter groups of types and . We compute the signed (by length)
generating function of this statistic over the whole group and over its maximal
and some other quotients and show that it always factors nicely. We also
present some conjectures
A symmetric unimodal decomposition of the derangement polynomial of type
The derangement polynomial for the symmetric group enumerates
derangements by the number of excedances. The derangement polynomial
for the hyperoctahedral group is a natural type analogue. A new
combinatorial formula for this polynomial is given in this paper. This formula
implies that 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 as the local
-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 . The
various decomposing polynomials introduced here are also studied in terms of
recurrences, generating functions, combinatorial interpretations, expansions
and real-rootedness.Comment: Changes in Remark 7.3 and the bibliograph