A New Statistic on the Hyperoctahedral Groups

Stasinski, Alexander; Voll, Christopher

Abstract

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank. For several descent classes we prove the conjectural formula. For this we construct suitable supporting sets for the relevant generating functions. We prove cancellations on the complements of these supporting sets using suitably defined sign reversing involutions.

Citation

Stasinski, A., & Voll, C. (2013). A New Statistic on the Hyperoctahedral Groups. Electronic Journal of Combinatorics, 20(3), Article 50

• Journal Article Type: Article
• Publication Date: Sep 26, 2013
• Deposit Date: May 2, 2014
• Publicly Available Date: May 6, 2014
• Journal: Electronic Journal of Combinatorics
• Publisher: Electronic Journal of Combinatorics
• Peer Reviewed: Peer Reviewed
• Volume: 20
• Issue: 3
• Article Number: 50
• Keywords: Hyperoctahedral groups, Signed permutation statistics, Sign reversing involutions, Descent sets, Generating functions.
• Publisher URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i3p50