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

我们提出一种统一的方法，通过专门研究Poirier的有色拟对称函数，来证明欧拉统计量和Mahonian统计量在一组有色排列上的联合分布的一般公式。我们采用这种方法来推导有关彩色排列，排列和对合的欧拉-马洪尼亚分布的公式。作为我们的结果的特例，已恢复了许多已知公式，包括Biagioli–Zeng，Assaf，Haglund–Loehr–Remmel，Chow–Mansour，Biagioli–Caselli，Bagno–Biagioli，Faliharimalala–Zeng的公式。还获得了几个新结果。例如，引入了有符号排列的两参数标志主索引，并证明了其分布以及与一些欧拉伙伴的联合分布的公式。