On a conjecture concerning the $r$-Euler-Mahonian statistic on permutations

Abstract

A pair $(\mathrm{s}{\mathrm{t}}_1,\mathrm{s}{\mathrm{t}}_2)$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{s}{\mathrm{t}}_1,\mathrm{s}{\mathrm{t}}_2)$ and $(\mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set ${\mathfrak{S}}_n$ of all permutations of $\{1,2,\dots ,n\}$, where $\mathrm{rdes}$ denotes the $r$-descent number and $\mathrm{rmaj}$ denotes the $r$-major index introduced by Rawlings. The main objective of this paper is to prove that $({\mathrm{exc}}_r,{\mathrm{den}}_r)$ and $(\mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over ${\mathfrak{S}}_n$, thereby confirming a recent conjecture posed by Liu. When $r=1$, the result recovers the equidistribution of $(\mathrm{des},\mathrm{maj})$ and $(\mathrm{exc},\mathrm{den})$, which was first conjectured by Denert and proved by Foata and Zeilberger.