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 $A$ and $B$. 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.

[1]  Sivaramakrishnan Sivasubramanian.  Signed excedance enumeration via determinants , 2011, Adv. Appl. Math..

[2]  Stasinski and Voll's Hyperoctahedral Group Conjecture , 2014 .

[3]  C. Voll,et al.  Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B , 2011, 1104.1756.

[4]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .

[5]  Jacques Désarménien,et al.  The signed Eulerian numbers , 1992, Discret. Math..

[6]  Victor Reiner.  Descents and one-dimensional characters for classical Weyl groups , 1995, Discret. Math..

[7]  Christopher Voll,et al.  A New Statistic on the Hyperoctahedral Groups , 2013, Electron. J. Comb..

[8]  Igusa-type functions associated to finite formed spaces and their functional equations , 2006, math/0603565.

[9]  A. Björner,et al.  Combinatorics of Coxeter Groups , 2005 .

[10]  Michelle L. Wachs,et al.  An involution for signed Eulerian numbers , 1992, Discret. Math..

[11]  Riccardo Biagioli.  Signed Mahonian polynomials for classical Weyl groups , 2006, Eur. J. Comb..

[12]  Ira M. Gessel,et al.  Signed Mahonians , 2005, J. Comb. Theory, Ser. A.

[13]  Angela Carnevale,et al.  Proof of a conjecture of Klopsch-Voll on Weyl groups of type $A$ , 2017, 1707.01002.

[14]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[15]  Fabrizio Caselli.  Signed mahonians on some trees and parabolic quotients , 2012, J. Comb. Theory, Ser. A.

[16]  Angela Carnevale,et al.  Odd length in Weyl groups , 2017, Algebraic Combinatorics.

[17]  Pietro Mongelli,et al.  Signed excedance enumeration in classical and affine Weyl groups , 2015, J. Comb. Theory, Ser. A.