Specializations of colored quasisymmetric functions and Euler–Mahonian identities

We propose a unified approach to prove general formulas for the joint distribution of an Eulerian and Mahonian statistic over set colored permutations by specializing Poirier's quasisymmetric functions. apply this method derive Euler–Mahonian distributions on permutations, derangements involutions. A number known are recovered as special cases our results, including Biagioli–Zeng, Assaf, Haglund–Loehr–Remmel, Chow–Mansour, Biagioli–Caselli, Bagno–Biagioli, Faliharimalala–Zeng. Several new results also obtained. For instance, two-parameter flag major index signed is introduced its with some partners proven.