  A pair $(\mathrm{st_1}, \mathrm{st_2})$ of permutation statistics is said to
be $r$-Euler-Mahonian if $(\mathrm{st_1}, \mathrm{st_2})$ and
  $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set
$\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$, 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.
