Euler-Mahonian Statistics On Ordered Set Partitions (II)

TL;DR

This paper investigates Euler-Mahonian statistics on ordered set partitions, providing bijective proofs for conjectures related to $p,q$-Stirling numbers and exploring combinatorial path diagram encodings.

Contribution

It offers the first bijective proofs of all conjectures by ein, connecting ordered set partitions with path diagrams and extending MacMahon's theorem to partitions.

Findings

Bijective proofs of all ein conjectures.

Path diagram encodings of ordered partitions.

Partition version of MacMahon's theorem.

Abstract

We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of \stein (Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahon's theorem on the equidistribution of the statistics inversion number and major index on words.