Fuss-Schr\"oder Paths and Rooted Plane Forests

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.