Mahonian and Euler-Mahonian statistics for set partitions

Abstract

A partition of the set $[n]:=\{1,2,\dots ,n\}$ 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 $B_1,B_2,\dots ,B_m$ satisfying $\max B_1<\max B_2<\cdots <\max B_m$, we represent it by a word $w=w_1{w}_2\dots w_n$ such that $i\in B_{w_i}$, $1\le i\le n$. 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.

References in corpus (1)

• An Extension of the Foata Map to Standard Young Tableaux