Fuss-Schr\"oder Paths and Rooted Plane Forests

TL;DR

This paper establishes a bijection between Fuss-Schr"oder paths and rooted plane forests, enabling enumeration of these paths and their generalizations, thus solving an open combinatorial problem and extending the concept to broader classes.

Contribution

It introduces a bijection between Fuss-Schr"oder paths and rooted plane forests, providing a recursive enumeration method and generalizing the paths to arbitrary sets of parameters.

Findings

Derived a recursion for counting large Fuss-Schr"oder paths

Solved an open enumeration problem by An, Jung, and Kim

Generalized paths to include arbitrary parameter sets

Abstract

We describe a bijection between $(k,k)$-Fuss-Schr\"oder paths of type $\lambda$ and certain rooted plane forests with $n(k+1)+2$ vertices. This yields a recursion which allows us to analytically enumerate the number of large $(k,r)$-Fuss-Schr\"oder paths of type $\lambda$, solving an open question posed by An, Jung, and Kim. Furthermore, we generalize the concept of $(k,r)$-Fuss-Schr\"oder paths to $(k,S)$-Fuss-Schr\"oder paths, in which $r$ can take any value in a given set $S$, and enumerate these paths as well.