Further results on r-Euler-Mahonian statistics

Abstract

As natural generalizations of the descent number ($\mathrm{des}$) and the major index ($\mathrm{maj}$), Rawlings introduced the notions of the r-descent number ($r\mathrm{des}$) and the r-major index ($r\mathrm{maj}$) for a given positive integer r. A pair $({\mathrm{st}}_1,{\mathrm{st}}_2)$ of permutation statistics is said to be r-Euler-Mahonian if $(s{t}_1,s{t}_2)$ and $(r\mathrm{des},r\mathrm{maj})$ are equidistributed over the set $S_n$ 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 $(g{\mathrm{exc}}_{\ell },g{\mathrm{den}}_{\ell })$ is $(g+\ell -1)$-Euler-Mahonian for all positive integers g and ℓ, where $g{\mathrm{exc}}_{\ell }$ denotes the g-gap ℓ-level excedance number and $g{\mathrm{den}}_{\ell }$ denotes the g-gap ℓ-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of $(g{\mathrm{exc}}_{\ell },g{\mathrm{den}}_{\ell })$ and $(r\mathrm{des},r\mathrm{maj})$ where $r=g+\ell -1$. Setting $g=\ell =1$, our 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. Our second main result is concerned with the analogous result for $(g{\mathrm{exc}}_{\ell },g{\mathrm{den}}_{g+\ell })$ which states that $(g{\mathrm{exc}}_{\ell },g{\mathrm{den}}_{g+\ell })$ is $(g+\ell -1)$-Euler-Mahonian for all positive integers g and ℓ.