Enumerating Wreath Products Via Garsia-Gessel Bijections

TL;DR

This paper extends Garsia-Gessel bijections to compute generating functions for complex permutation statistics over wreath products, unifying multiple known identities across Coxeter groups of types A and B.

Contribution

It generalizes and unifies existing identities by developing new bijections for wreath products involving symmetric and cyclic groups.

Findings

Derived formulas for generating functions of permutation statistics.

Unified several known identities in Coxeter group theory.

Enhanced understanding of permutation statistics over wreath products.

Abstract

We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics $(\des_G, \maj,\ell_G, \col)$ and $(\des_G, \ides_G, \maj, \imaj, \col, \icol)$ over the wreath product of a symmetric group by a cyclic group. Here $\des_G$, $\ell_G$, $\maj$, $\col$, $\ides_G$, $\imaj_G$, and $\icol$ denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type $A$ and $B$.