Generalizations of wreath product identities via Garsia-Gessel bijections

TL;DR

This paper generalizes and simplifies multivariate generating function identities for permutation statistics related to wreath products, extending previous results to broader orderings and providing more unified formulas.

Contribution

It extends Garsia-Gessel bijections to general positive dominant orderings and simplifies existing multivariate distribution functions for wreath products.

Findings

Generalized four-variate identities to any positive dominant ordering

Simplified six-variate distribution functions under Adin-Roichman ordering

Unified framework for wreath product permutation statistics

Abstract

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.