Generalized Catalan Numbers and the Enumeration of Planar Embeddings

TL;DR

This paper introduces a new combinatorial interpretation of Raney numbers using planar embeddings of trees, generalizing Catalan numbers, and explores their identities and applications to web enumeration.

Contribution

It provides a novel combinatorial interpretation of Raney numbers via planar tree embeddings, extending Catalan number concepts and linking to web enumeration.

Findings

New interpretation of Raney numbers in terms of planar embeddings

Derivation of combinatorial identities involving Raney numbers

Identification of specific Raney numbers with oriented trees satisfying certain properties

Abstract

The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all 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 $R_{p,1}(n) =_{p} c_n$. 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 $R_{p^2,p}(n)$ with oriented trees whose vertices satisfy the "source or sink property". We close with comments applying these results to the enumeration of connected (non-elliptic) $A_2$ webs that lack an internal cycle.