Recently, we proved the equidistribution of the pairs of permutation statistics (rdes, rmaj) and (rexc, rden). Any pair of permutation statistics that is equidistributed with these pairs is said to be r-Euler--Mahonian. Several classes of r-Euler--Mahonian statistics were established by Huang--Lin--Yan and Huang--Yan. Inspired by their bijections, we provide a new bijective proof of the classical result that (exc, den) is Euler--Mahonian. Using this bijection, we further show that (excₑ, den) is r-Euler--Mahonian, where excₑ denotes the number of r-level excedances (i. e. , excedances at least r). Furthermore, by extending our bijection, we establish a more general result that encompasses all the aforementioned results.
Shao-Hua Liu (Mon,) studied this question.