Mahonian and Euler-Mahonian statistics for set partitions

Abstract

A partition of the set [n]:={1,2,…,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 B1,B2,…,Bm satisfying max⁡B1<max⁡B2<⋯<max⁡Bm, we represent it by a word w=w1w2…wn such that i∈Bwi, 1≤i≤n. We prove that the Mahonian statistics INV, MAJ, MAJd, 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.