Authors: Masao Ishikawa ¹; Anisse Kasraoui ²; Jiang Zeng ²

• 1 Tottori University
• 2 Institut Camille Jordan

An ordered partition of $[n]:=\{1,2,\ldots, n\}$ 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.