Statistics on Ordered Partitions of Sets and q-Stirling Numbers

An ordered partition of [n]:={1,2,..., n} is a sequence of its disjoint subsets whose union is [n]. The number of ordered partitions of [n] with k blocks is k!S(n,k), where S(n,k) is the Stirling number of second kind. In this paper we prove some refinements of this formula by showing that the generating function of some statistics on the set of ordered partitions of [n] with k blocks is a natural $q$-analogue of k!S(n,k). In particular, we prove several conjectures of Steingr\'{\i}msson. To this end, we construct a mapping from ordered partitions to walks in some digraphs and then, thanks to transfer-matrix method, we determine the corresponding generating functions by determinantal computations.