Further results on $r$-Euler-Mahonian statistics

As natural generalizations of the descent number ($\text{des}$) and the major index ($\text{maj}$), Rawlings introduced the notions of the $r$-descent number ($r\text{des}$) and the $r$-major index ($r\text{maj}$) for a given positive integer $r$. A pair $({\text{st}}_1,{\text{st}}_2)$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{s}{\mathrm{t}}_1,\mathrm{s}{\mathrm{t}}_2)$ and $(r\text{des},r\text{maj})$ are equidistributed over the set ${\mathfrak{S}}_n$ of all permutations of $\{1,2,\dots ,n\}$. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that $(g{\text{exc}}_{\mathrm{\ell }},g{\text{den}}_{\mathrm{\ell }})$ is $(g+\mathrm{\ell }-1)$-Euler-Mahonian for all positive integers $g$ and $\mathrm{\ell }$, where $g{\text{exc}}_{\mathrm{\ell }}$ denotes the $g$-gap $\mathrm{\ell }$-level excedance number and $g{\text{den}}_{\mathrm{\ell }}$ denotes the $g$-gap $\mathrm{\ell }$-level Denert’s statistic. This is accomplished via a bijective proof of the equidistribution of $(g{\text{exc}}_{\mathrm{\ell }},g{\text{den}}_{\mathrm{\ell }})$ and $(r\text{des},r\text{maj})$ where $r=g+\mathrm{\ell }-1$. Setting $g=\mathrm{\ell }=1$, our result recovers the equidistribution of $(\text{des},\text{maj})$ and $(\text{exc},\text{den})$, which was first conjectured by Denert and proved by Foata and Zeilberger. Our second main result is concerned with the analogous result for $(g{\text{exc}}_{\mathrm{\ell }},g{\text{den}}_{g+\mathrm{\ell }})$ which states that $(g{\text{exc}}_{\mathrm{\ell }},g{\text{den}}_{g+\mathrm{\ell }})$ is $(g+\mathrm{\ell }-1)$-Euler-Mahonian for all positive integers $g$ and $\mathrm{\ell }$.