Recently, we proved the equidistribution of the pairs of permutation statistics $(r\textsf{ des},r\textsf{ maj})$ and $(r\textsf{ exc},r\textsf{ den})$. 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 $(\textsf{ exc},\textsf{ den})$ is Euler--Mahonian. Using this bijection, we further show that $(\textsf{ exc}_{ r},\textsf{ den})$ is $r$-Euler--Mahonian, where $\textsf{ exc}_{ r}$ 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.
Categories: math.CO