Euler-mahonian Statistics on Ordered 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 Euler-Mahonian statistics if its generating polynomial is [k]q!Sq(n, k), which is a natural q-analogue of k!S(n, k). Motivated by Steingŕımsson’s conjectures dated back to 1997, we consider two different methods to produce Euler-Mahonian statistics on ordered partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of Françon-Viennot’s bijection to derive many Euler-Mahonian statistics by expanding the generating function of [k]q!Sq(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 Steingŕımsson.