Restricted partitions and generalized Catalan numbers

The generalized Catalan numbers w n are given by the recurrence w n =2w n-1 +∑ i=1 n-2 w i w n-2-i if n≥2, with w 0 =w 1 =1, and count a restricted subset of the Catalan paths having semilength n. In this paper, we provide new combinatorial interpretations of these numbers in terms of finite set partitions. In particular, we identify five classes of the partitions of size n, all of which have cardinality w n and each avoiding a set of two classical patterns of length four. We use both combinatorial and algebraic arguments to establish our results, applying the kernel method in a couple of the apparently more difficult cases.