Generalized Catalan numbers from hypergraphs

TL;DR

This paper introduces a new family of generalized Catalan numbers derived from hypergraph-based matrix models, expanding combinatorial interpretations and proposing asymptotic conjectures.

Contribution

It defines an infinite collection of generalized Catalan numbers from hypergraph matrix models, extending classical combinatorial sequences.

Findings

Defined the sequence C_n^(m) for m >= 1

Provided combinatorial interpretations of the generalized numbers

Conjectured asymptotic behaviors of the sequences

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.