  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.
