On some analogues of Carlitz's identity for the hyperoctahedral group

TL;DR

This paper connects and generalizes identities related to the hyperoctahedral group by linking different major indices and computing multivariate generating series for various Coxeter groups.

Contribution

It introduces a new description of the flag major index and generalizes key identities to Coxeter groups of types B and D.

Findings

Established a connection between Reiner's and Adin-Roichman's major indices.

Generalized Chow and Gessel's identity to type B and D Coxeter groups.

Computed multivariate generating series for descents, major index, length, and negative entries.

Abstract

We give a new description of the flag major index, introduced by Adin and Roichman, by using a major index defined by Reiner. This allows us to establish a connection between an identity of Reiner and some more recent results due to Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel by computing the four-variate generating series of descents, major index, length, and number of negative entries over Coxeter groups of type $B$ and $D$.