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arXiv · 2307.05198

An Inversion Statistic on the Hyperoctahedral Group

Hasan Arslan·Alnour Altoum·Hilal Karakus Arslan
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Abstract

In this paper, we introduce an inversion statistic on the hyperoctahedral group $B_n$ by using an decomposition of a positive root system of this reflection group. Then we prove some combinatorial properties for the inversion statistic. We establish an enumeration system on the group $B_n$ and give an efficient method to uniquely derive any group element known its enumeration order with the help of the inversion table. In addition, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on $B_n$.

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Hasan Arslan, Alnour Altoum, Hilal Karakus Arslan. 2023-07-20. An Inversion Statistic on the Hyperoctahedral Group. https://arxiv.org/abs/2307.05198

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