An Inversion Statistic on the Hyperoctahedral Group

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 $\mathit{\text{flag-major}}\mathit{\text{index}}$ is equi-distributed with this inversion statistic on $B_n$.