Signed Mahonians

@article{Adin2005SignedM,
  title={Signed Mahonians},
  author={Ron M. Adin and Ira M. Gessel and Yuval Roichman},
  journal={J. Comb. Theory, Ser. A},
  year={2005},
  volume={109},
  pages={25-43}
}
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.