r-Euler-Mahonian statistics on permutations

Abstract

Let $r\text{des}$ and $r\text{exc}$ denote the permutation statistics r-descent number and r-excedance number, respectively. We prove that the pairs of permutation statistics $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$ are equidistributed, where $r\text{maj}$ denotes the r-major index defined by Don Rawlings and $r\text{den}$ denotes the r-Denert's statistic defined by Guo-Niu Han. When $r=1$, this result reduces to the equidistribution of $(\text{des},\text{maj})$ and $(\text{exc},\text{den})$, which was conjectured by Denert in 1990 and proved that same year by Foata and Zeilberger. We call a pair of permutation statistics that is equidistributed with $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$ an r-Euler-Mahonian statistic, which reduces to the classical Euler-Mahonian statistic when $r=1$.

We then introduce the notions of r-level descent number, r-level excedance number, r-level major index, and r-level Denert's statistic, denoted by ${\text{des}}_r,{\text{exc}}_r,{\text{maj}}_r$, and ${\text{den}}_r$, respectively. We prove that $({\text{des}}_r,{\text{maj}}_r)$ is r-Euler-Mahonian and conjecture that $({\text{exc}}_r,{\text{den}}_r)$ is r-Euler-Mahonian. Furthermore, we give an extension of the above result and conjecture.