Euler-Mahonian distributions of type $B_n$

TL;DR

This paper establishes a bijective proof for the equidistribution of certain Euler-Mahonian statistics over the hyperoctahedral group, providing new insights and a novel proof of the generalized Carlitz identity.

Contribution

It constructs a bijection demonstrating the equidistribution of specific statistics and offers a new proof of the generalized Carlitz identity for type B.

Findings

Bijection confirming equidistribution of statistics

New proof of the generalized Carlitz identity

Enhanced understanding of Euler-Mahonian distributions in type B

Abstract

Adin, Brenti, and Roichman introduced the pairs of statistics $(\ndes, \nmaj)$ and $(\fdes, \fmaj)$. They showed that these pairs are equidistributed over the hyperoctahedral group $B_n$, and can be considered "Euler-Mahonian" in that they generalize the Carlitz identity. Further, they asked whether there exists a bijective proof of the equidistribution of their statistics. We give such a bijection, along with a new proof of the generalized Carlitz identity.