zbMATH — the first resource for mathematics

Odd length for even hyperoctahedral groups and signed generating functions. (English) Zbl 1370.05010
Summary: 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.

05A15 Exact enumeration problems, generating functions
05A05 Permutations, words, matrices
11M41 Other Dirichlet series and zeta functions
20F55 Reflection and Coxeter groups (group-theoretic aspects)
Full Text: DOI arXiv
[1] Adin, R.; Gessel, I.; Roichman, Y., Signed mahonians, J. Combin. Theory Ser. A, 109, 25-43, (2005) · Zbl 1059.05002
[2] Biagioli, R., Signed Mahonian polynomials for classical Weyl groups, European J. Combin., 27, 207-217, (2006) · Zbl 1082.05003
[3] Björner, A.; Brenti, F., (Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231, (2005), Springer-Verlag New York) · Zbl 1110.05001
[4] Brenti, F.; Carnevale, A., Proof of a conjecture of klopsch-voll on Weyl groups of type \(A\), Trans. Amer. Math. Soc., 369, 10, 7531-7547, (2017) · Zbl 1368.05007
[5] Carnevale, A., Odd Length in Weyl Groups: Proof of Two Conjectures and Properties, (2015), Università di Roma Tor Vergata, (Ph.D. thesis)
[6] Caselli, F., Signed mahonians on some trees and parabolic quotients, J. Combin. Theory Ser. A, 119, 1447-1460, (2012) · Zbl 1245.05004
[7] Désarménien, J.; Foata, D., The signed Eulerian numbers, Discrete Math., 99, 49-58, (1992) · Zbl 0769.05094
[8] Klopsch, B.; Voll, C., Igusa-type functions associated to finite formed spaces and their functional equations, Trans. Amer. Math. Soc., 361, 8, 4405-4436, (2009) · Zbl 1229.05288
[9] A. Landesman, Stasinski and Voll’s Hyperoctahedral Group Conjecture, arXiv:1408.7105.
[10] Mantaci, R., Statistiques Euleriennes sur les Groupes de Permutation, (1991), Université Paris, (Ph.D. thesis)
[11] Mongelli, P., Signed excedance enumeration in classical and affine Weyl groups, J. Combin. Theory Ser. A, 130, 129-149, (2015) · Zbl 1303.05003
[12] Reiner, V., Descents and one-dimensional characters for classical Weyl groups, Discrete Math., 140, 129-140, (1995) · Zbl 0830.20025
[13] Sivasubramanian, S., Signed excedance enumeration via determinants, Adv. Appl. Math., 47, 783-794, (2011) · Zbl 1233.05029
[14] Stasinski, A.; Voll, C., A new statistic on the hyperoctahedral groups, Electron. J. Combin., 20, 3, (2013), Paper 50, 23 pp · Zbl 1295.05038
[15] Stasinski, A.; Voll, C., Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B, Amer. J. Math., 136, 2, 501-550, (2014) · Zbl 1286.11140
[16] Wachs, M., An involution for signed Eulerian numbers, Discrete Math., 99, 59-62, (1992) · Zbl 0769.05098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.