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Proof of Stasinski and Voll’s hyperoctahedral group conjecture. (English) Zbl 1406.05010
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.

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
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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
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[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
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