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 $λ$.

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