Counting Pattern-free Set Partitions I: A Generalization of Stirling Numbers of the Second Kind

A partition u of [ k ] = {1, 2,⋯ , k } is contained in another partition v of [ l ] if [ l ] has a k -subset on which v inducesu . We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of the Stanley–Wilf conjecture is proposed. We prove that the generating function (GF) counting v is rational if (i)R is finite and the number of parts of v is fixed or if (ii)u has only singleton parts and at most one doubleton part. In fact, (ii) is an application of (i). As another application of (i) we prove that for each k the GF counting partitions with k pairs of crossing parts belongs toZ(1 − 4 x).