A combinatorial interpretation of Mahonian numbers of type B

In this paper, we first introduce the number of signed permutations with exactly $k$ inversions, which is denoted by $i_B(n,k)$ and called \textit{Mahonian numbers of type $B$}. Then we provide a recurrence relation for the Mahonian numbers $i_B(n,k)$. In addition, we give an explicit recursive description for the summation of inversions of all permutations in the hyperoctahedral group $B_n$, denoted by $\mathcal{B}_n$. Furthermore, we enumerate the total number of inversions in permutations in the hyperoctahedral group $B_n$ concretely with the help of an inversion statistic and a backward permutation concepts on $B_n$.