Abstract
The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.
Highlights
The Catalan numbers (Cn)n 01, 1, 2, 5, 14, 42, 132, 429, 1430, . . . , form one of the most venerable sequences in combinatorics
Cn counts the number of plane trees with n + 1 vertices
Cn counts the number of Dyck paths of length 2n
Summary
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . , form one of the most venerable sequences in combinatorics. Cn counts the number of ballot sequences of length 2n. Cn counts the number of binary plane trees with n vertices. Cn counts the number of pairings of the sides of Π such that Σπ is orientable and has genus 0, i.e. is homeomorphic to the 2-sphere. A reference for combinatorial interpretations of Catalan numbers is Richard Stanley’s recent monograph [17]. It is seen to be equivalent to [17, Ch. 2, (59)], which counts the number of ways to draw n nonintersecting chords joining 2n points on the circumference of a circle. Another resource is OEIS [14], where the Catalans are sequence A000108.
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