Fuss-Schr\"oder Paths and Rooted Plane Forests
Michael Kural

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 -Fuss-Schr\"oder paths of type and certain rooted plane forests with vertices. This yields a recursion which allows us to analytically enumerate the number of large -Fuss-Schr\"oder paths of type , solving an open question posed by An, Jung, and Kim. Furthermore, we generalize the concept of -Fuss-Schr\"oder paths to -Fuss-Schr\"oder paths, in which can take any value in a given set , and enumerate these paths as well.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Polynomial and algebraic computation · Topological and Geometric Data Analysis
