Landesman, Aaron Proof of Stasinski and Voll’s hyperoctahedral group conjecture. (English) Zbl 1406.05010 Australas. J. Comb. 71, Part 2, 196-240 (2018). Summary: In a recent paper, A. Stasinski and C. Voll [Electron. J. Comb. 20, No. 3, Research Paper P50, 23 p. (2013; Zbl 1295.05038)] introduced a length-like statistic on hyperoctahedral groups and conjectured a product formula for this statistic’s signed distribution over arbitrary quotients. Stasinski and Voll [loc. cit.] proved this conjecture for a few special types of quotients. We prove this conjecture in full, showing it holds for all quotients. In the case of signed permutations with at most one descent, this formula gives the Poincaré polynomials for the varieties of symmetric matrices of a fixed rank. Cited in 3 Documents MSC: 05A15 Exact enumeration problems, generating functions 05A05 Permutations, words, matrices 05E15 Combinatorial aspects of groups and algebras (MSC2010) 11M41 Other Dirichlet series and zeta functions Keywords:Poincaré polynomials; hyperoctahedral groups; signed permutations PDF BibTeX XML Cite \textit{A. Landesman}, Australas. J. Comb. 71, Part 2, 196--240 (2018; Zbl 1406.05010) Full Text: Link arXiv References: [1] A. Bj¨orner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics Vol. 231, Springer, New York, 2005. [2] F. Brenti and A. Carnevale, Proof of a conjecture of Klopsch-Voll on Weyl groups of type A, Trans. Amer. Math. Soc. 369(10) (2017), 7531–7547. · Zbl 1368.05007 [3] B. Klopsch and C. Voll, Igusa-type functions associated to finite formed spaces and their functional equations, Trans. Amer. Math. Soc. 361(8) (2009), 4405– 4436. · Zbl 1229.05288 [4] A. Stasinski and C. Voll, A new statistic on the hyperoctahedral groups, Electron. J. Combin. 20(3) (2013), Paper 50, 23pp. · Zbl 1295.05038 [5] A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B, Amer. J. Math. 136(2) (2014), 501–550. · Zbl 1286.11140 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.