Injectively $k$-colored rooted forests

TL;DR

This paper develops enumeration formulas for injectively k-colored rooted forests with specified vertex and root color distributions, leading to new insights into Fuss-Catalan number distributions and polygon triangulations.

Contribution

It introduces novel enumeration methods for injectively k-colored rooted forests and applies these to derive new distributions of Fuss-Catalan numbers and polygon triangulation counts.

Findings

Derived enumeration formulas for injectively k-colored rooted forests.

Established new multi-parameter distributions of Fuss-Catalan numbers.

Enumerated polygon triangulations based on proper 3-colorings.

Abstract

We enumerate injectively $k$-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an additional application we enumerate triangulations of regular convex polygons according to their proper 3-coloring type.