Generating Functions of Some Families of Directed Uniform Hypergraphs

TL;DR

This paper develops generating functions to count specific families of directed uniform hypergraphs, extending known results on acyclic and strongly connected directed graphs.

Contribution

It introduces a new generating function framework for counting acyclic and strongly connected uniform directed hypergraphs, generalizing existing graph enumeration results.

Findings

Derived explicit formulas for counting acyclic hypergraphs.

Extended enumeration techniques to strongly connected hypergraphs.

Unified approach to counting directed hypergraph structures.

Abstract

In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of directed acyclic graphs and the number of strongly connected directed graphs.