Generalizations of wreath product identities via Garsia-Gessel bijections

Garsia and Gessel constructed innovative bijections to obtain multivariate generating functions of permutation statistics. In 2011, Biaogioli and Zeng successfully derived four and six variate distributions on the set of wreath product. In this paper, we will generalize the four variate identities from BZ to any positive dominant ordering. And we will simplify the six variate distribution function under the ordering originally defined by Adin and Roichman in 2001.