Restricted partitions and generalized Catalan numbers

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)239–251
JournalPure Mathematics and Applications
Issue number2
StatePublished - 1 Jan 2011


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