Generalized Catalan numbers from hypergraphs

Paul E. Gunnells¹

¹UMass Amherst

The electronic journal of combinatorics, Tome 28 (2021) no. 1

Résumé

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.

Détail

DOI : 10.37236/8733

Classification : 05A10, 05A18, 11B65
Mots-clés : walks on trees, hypergraph Catalan numbers