  The Catalan numbers (C_n)_{n >= 0} = 1,1,2,5,14,42,... 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 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 define an infinite collection of generalizations C_n^(m), m
>= 1, with m=1 giving the usual Catalans. The sequence C_n^(m) comes from
studying certain matrix models attached to hypergraphs. We also give some
combinatorial interpretations of these numbers, and conjecture some
asymptotics.
