Signed Mahonians
  A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type $B$ analogues. 
Ron M. Adin, Ira M. Gessel and Yuval Roichman