Generalized Catalan numbers and the enumeration of planar embeddings

The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns. We give a new combinatorial interpretation for the Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the p-Catalan numbers in terms of p-ary trees via the specialization Rp,1(k) = pck. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of Rp2,p(k) with oriented trees whose vertices satisfy the “source-sink property.” We close with comments applying these results to the enumeration of connected (non-elliptic) A2 webs that lack an internal cycle.