The Garsia–Gessel bijection in known to be a powerful tool in deriving multivariate generating functions. In 2011, Biagioli and Zeng first introduced an ordering, now known as the Biagioli–Zeng partial order, and successfully used the Garsia–Gessel bijection to obtain four-variable and six-variable generating functions for the wreath product Zr≀Sn. However, the construction of these generating functions relies heavily on the specific definition of the Biagioli–Zeng partial order, and their resulting forms are rather complicated and difficult to interpret.

In this paper, we unify several existing partial orders on wreath product, including the Biagioli–Zeng partial order, within a more general framework called the positive-dominant ordering. Then we prove that the four-variable generating function of Biagioli–Zeng can be successfully generalized to arbitrary positive-dominant orderings, and that their six-variable generating function can be significantly simplified, yielding a formulation that aligns more naturally with classical multivariate generating functions on traditional Sn, under A-R’s ordering which was originally defined by Adin and Roichman in 2001.

Dr. Tingyao Xiong, is currently an Associate Professor at Radford University, U.S.A. She got her Bachelor’s Degree in Mathematics from Nanjing University in China. She continued her study in Mathematics and Statistics at Michigan State University, where she got her Ph.D. degree under the supervision of Professor Jonathan Hall. Dr. Xiong’s Ph.D. dissertation was funded by U.S. NSF with award # 0601621. Dr. Xiong is an active scholar with wide research interest in area of Number Theory, Combinatorics, Information Theory and Financial Mathematics. She has given presentations on various international conferences including AMS Joint Meeting and IEEE ISIT.