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 iB(n,k) and called Mahonian numbers of type B. Then we provide a recurrence relation for the Mahonian numbers iB(n,k). In addition, we give an explicit recursive description for the summation of inversions of all permutations in the hyperoctahedral group Bₙ, denoted by Bₙ. Furthermore, we enumerate the total number of inversions in permutations in the hyperoctahedral group Bₙ concretely with the help of an inversion statistic and a backward permutation concepts on Bₙ.