On the descent numbers and major indices for the hyperoctahedral group

Adin, Brenti, and Roichman [R.M. Adin, F. Brenti, Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. 27 (2001) 210–224], in answering a question posed by Foata, introduced two descent numbers and major indices for the hyperoctahedral group Bn, whose joint distribution generalizes an identity due to MacMahon and Carlitz. We shall show that yet another pair of statistics exists, and whose joint distribution constitutes a “natural” solution to Foata's problem.