Enumeration of Fuss-Schröder paths

In this paper we enumerate the number of (k, r)-Fuss-Schröder paths of type λ. Y. Park and S. Kim studied small Schröder paths with type λ. Generalizing the results to small (k, r)-Fuss-Schröder paths with type λ, we give a combinatorial interpretation for the number of small (k, r)-Fuss-Schröder paths of type λ 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 λ.