New r-Euler--Mahonian statistics involving Denert's statistic

2025/08/18 by Shao-Hua Liu, Liu, Shao-Hua
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2508.12717

Abstract

Recently, we proved the equidistribution of the pairs of permutation statistics (r\textsfdes,r\textsfmaj) and (r\textsfexc,r\textsfden). 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 (\textsfexc,\textsfden) is Euler--Mahonian. Using this bijection, we further show that (\textsfexc_r,\textsfden) is r-Euler--Mahonian, where \textsfexc_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.

Citations

• Further results on r-Euler-Mahonian statistics
• On a conjecture concerning the r-Euler-Mahonian statistic on permutations
• An extension of MacMahon's Equidistribution Theorem to ordered set partitions
• A classification of mahonian maj-inv statistics
• Théorie Géométrique des Polynômes Eulériens
• Denert's Permutation Statistic Is Indeed Euler‐Mahonian
• Interprétations combinatoires des nombres de Genocchi

Related

• On a conjecture concerning the r-Euler-Mahonian statistic on permutations
• Further results on r-Euler-Mahonian statistics
• On the distribution of some Euler-Mahonian statistics
• Denert's Permutation Statistic Is Indeed Euler‐Mahonian
• Euler-Mahonian distributions of type B_n