2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76081A classical result of MacMahon shows that the length function and the major index are equi-distributed over the symmetric group. Foata and Schützenberger 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.23 pp., to appear in J. Combin. Theory (Ser. A); reference added in proofCombinatoricsGroup Theorytext