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arXiv · 2102.05121

Generalized Catalan numbers from hypergraphs

Paul E. Gunnells
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Abstract

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.

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Paul E. Gunnells. 2021-02-09. Generalized Catalan numbers from hypergraphs. https://arxiv.org/abs/2102.05121

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