Counting acyclic and strong digraphs by descents

TL;DR

This paper introduces new generating functions to count various classes of directed graphs, including acyclic and strong digraphs, based on the concept of descents, providing a novel combinatorial approach.

Contribution

It develops a new type of generating function called graphic Eulerian generating functions to count strong and acyclic digraphs by descents, extending existing enumeration methods.

Findings

Derived formulas for counting strong tournaments using Eulerian generating functions.

Introduced graphic Eulerian generating functions for counting strong and acyclic digraphs.

Provided enumeration results for various classes of directed graphs based on descents.

Abstract

A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.