Hypergraph Fuss-Catalan Numbers

TL;DR

This paper introduces and explores the Hypergraph Fuss-Catalan numbers, generalizing classical combinatorial sequences, providing combinatorial interpretations and deriving their generating functions.

Contribution

It combines Fuss-Catalan and Hypergraph Catalan numbers, offering new combinatorial insights and explicit generating functions for these generalized numbers.

Findings

Provided combinatorial interpretations of Hypergraph Fuss-Catalan numbers.

Derived the generating function for these numbers.

Connected these numbers to existing combinatorial structures.

Abstract

The Catalan numbers $C_n$ are an extremely well-studied sequence of numbers that appear as the answer to many combinatorial problems. Two generalizations of these numbers that have been studied are the Fuss-Catalan numbers and the Hypergraph Catalan numbers. In this paper, we study the combination of these, the Hypergraph Fuss-Catalan numbers. We provide some combinatorial interpretations of these numbers, as well as describe their generating function.