  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
$B_n$.
