Equi-distribution over Descent Classes of the Hyperoctahedral Group

Abstract

A classical result of MacMahon shows that the length function and the major index are equi-distributed over the symmetric group. Foata and Sch\"utzenberger gave a remarkable refinement and proved that these parameters are equi-distributed over inverse descent classes, implying bivariate equi-distribution identities. Type $B$ analogues of these results, refinements and consequences are given in this paper.