Euler-Mahonian statistics on ordered set partitions

An ordered partition with k blocks of [n] := {1,2,...,n} is a sequence of k disjoint and nonempty subsets, called blocks, whose union is [n]. Clearly the number of such ordered partitions is k!S(n, k), where S(n,k) is the Stirling number of the second kind. A statistic on ordered partitions of [n] with k blocks is called an Euler-Mahonian statistic if its generating polynomial is [k]_q!S_q(n, k), which is a natural q-analogue of k!S(n, k). Motivated by Steingrfmsson's conjectures dating back to 1997, we consider two different methods to produce Euler-Mahonian statistics on ordered set partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of Francon-Viennot's bijection to derive many Euler-Mahonian statistics by expanding the generating function of [k] _q!S_q(n, k) as an explicit continued fraction; (b) we encode ordered partitions by walks in some digraphs and then derive new Euler-Mahonian statistics by computing their generating functions using the transfer-matrix method. In particular, we prove several conjectures of Steingrfmsson.