Enumeration of Fuss-Schröder paths

  • Suhyung An
  • JiYoon Jung
  • Sangwook Kim
Keywords: Fuss-Schröder paths, Type, Sparse noncrossing partitions


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 $\lambda$.
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