An ordered partition of [n]:={1,2,…, n} is a sequence of disjoint and nonempty subsets, called blocks, whose union is [n]. The aim of this paper is to compute some generating functions of ordered partitions by the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson, which assert that the generating function of some statistics of ordered partitions give rise to a natural q-analogue of k!S(n,k), where S(n,k) is the Stirling number of the second kind.