Mahonian and Euler-Mahonian statistics for set partitions

TL;DR

This paper introduces a novel representation of set partitions and demonstrates that various Mahonian and Euler-Mahonian statistics are equidistributed under this framework, revealing new combinatorial symmetries.

Contribution

It establishes the equidistribution of multiple Mahonian and Euler-Mahonian statistics for set partitions using a new word-based representation.

Findings

Mahonian statistics are equidistributed on set partitions

Euler-Mahonian statistics are also equidistributed

New representation links set partitions to permutation statistics

Abstract

A partition of the set $[n]:=\{1,2,\ldots,n\}$ is a collection of disjoint nonempty subsets (or blocks) of $[n]$, whose union is $[n]$. In this paper we consider the following rarely used representation for set partitions: given a partition of $[n]$ with blocks $B_{1},B_{2},\ldots,B_{m}$ satisfying $\max B_{1}<\max B_{2}<\cdots<\max B_{m}$, we represent it by a word $w=w_{1}w_{2}\ldots w_{n}$ such that $i\in B_{w_{i}}$, $1\leq i\leq n$. We prove that the Mahonian statistics INV, MAJ, MAJ$_{d}$, $r$-MAJ, Z, DEN, MAK, MAD are all equidistributed on set partitions via this representation, and that the Euler-Mahonian statistics (des, MAJ), (mstc, INV), (exc, DEN), (des, MAK) are all equidistributed on set partitions via this representation.