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

Abstract

A pair $(s{t}_1,s{t}_2)$ of permutation statistics is said to be r-Euler-Mahonian if $(s{t}_1,s{t}_2)$ and (rdes, rmaj) are equidistributed over the set $S_n$ of all permutations of $\{1,2,\dots ,n\}$, where rdes denotes the r-descent number and 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 (rdes, rmaj) are equidistributed over $S_n$, 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.