Further results on $r$-Euler-Mahonian statistics

TL;DR

This paper proves a conjecture relating generalized permutation statistics to $r$-Euler-Mahonian distributions, extending classical results and providing bijective proofs for their equidistribution.

Contribution

It confirms Liu's conjecture that certain $g$-gap level statistics are $r$-Euler-Mahonian, generalizing known results for classical permutation statistics.

Findings

Proved $(g ext{exc}_ ext{l}, g ext{den}_ ext{l})$ is $(g+ ext{l}-1)$-Euler-Mahonian.

Established bijective proof of equidistribution between these statistics and $(r ext{des}, r ext{maj})$.

Extended classical results to broader classes of permutation statistics.

Abstract

As natural generalizations of the descent number ($\des$) and the major index ($\maj$), Rawlings introduced the notions of the $r$-descent number ($r\des$) and the $r$-major index ($r\maj$) for a given positive integer $r$. A pair $(\st_1, \st_2)$ of permutation statistics is said to be $r$-Euler-Mahonian if $ (\mathrm{st_1}, \mathrm{st_2})$ and $ (r\des, r\maj)$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that $(g\exc_\ell, g\den_\ell)$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$, where $g\exc_\ell$ denotes the $g$-gap $\ell$-level excedance number and $g\den_\ell$ denotes the $g$-gap $\ell$-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of $(g\exc_\ell, g\den_\ell)$ and $ (r\des, r\maj)$ where $r=g+\ell-1$. Setting $g=\ell=1$, our result recovers the equidistribution of $(\des, \maj)$ and $(\exc, \den)$, which was first conjectured by Denert and proved by Foata and Zeilberger. Our second main result is concerned with the analogous result for $(g\exc_\ell, g\den_{g+\ell})$ which states that $(g\exc_\ell, g\den_{g+\ell})$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$.