Action graphs, planar rooted forests, and self-convolutions of the Catalan numbers

TL;DR

This paper establishes a connection between action graphs and planar rooted forests, demonstrating that certain graph families produce self-convolutions of Catalan numbers, with proofs based on combinatorial comparisons.

Contribution

It introduces a novel link between action graphs and Catalan number convolutions through planar rooted forests, expanding combinatorial understanding.

Findings

Action graphs generate self-convolutions of Catalan numbers.

Comparison with planar rooted forests proves the main result.

Provides a new combinatorial interpretation of Catalan convolutions.

Abstract

We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.