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Computing generating functions of ordered partitions with the transfer-matrix method

Abstract : 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.
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Masao Ishikawa, Anisse Kasraoui, Jiang Zeng. Computing generating functions of ordered partitions with the transfer-matrix method. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.193-202. ⟨hal-01184713⟩

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