Signed mahonians on some trees and parabolic quotients

We study the distribution of the major index with sign on some parabolic quotients of the symmetric group, extending and generalizing simultaneously results of Gessel–Simion and Adin–Gessel–Roichman, and on the labellings of some special trees that we call rakes. We further consider and compute the distribution of the flag-major index on some parabolic quotients of wreath products and other related groups. All these distributions turn out to have very simple factorization formulas.