Enumeration of Fuss-Schröder paths

Abstract

In this paper we enumerate the number of $(k,r)$-Fuss-Schröder paths of type $\lambda$. Y. Park and S. Kim studied small Schröder paths with type $\lambda$. Generalizing the results to small $(k,r)$-Fuss-Schröder paths with type $\lambda$, we give a combinatorial interpretation for the number of small $(k,r)$-Fuss-Schröder paths of type $\lambda$ by using Chung-Feller style. We also give two sets of sparse noncrossing partitions of $[2(k+1)n+1]$ and $[2(k+1)n+2]$ which are in bijection with the set of all small and large, respectively, $(k,r)$-Fuss-Schröder paths of type $\lambda$.