• 호남수학학술지
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• 제43권4호
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• pp.641-653
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• 2021

Armstrong enumerated the number of Fuss-Catalan paths with a given type and Rhoades provided the number of Dyck paths with a given type and a given number of blocks. In this paper we generalize those results to enumerate the number of Fuss-Catalan paths with a fixed type and a fixed number of blocks. We provide two proofs of this result. The first one uses the Chung-Feller theorem and a certain polynomial, while the second one is bijective. Also, we give a conjecture generalizing this result to the family of small Fuss-Schröder paths.

• 충청수학회지
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• 제35권4호
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• pp.335-343
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• 2022

We show that valleys, high peaks, and modular ascents are statistics of Fuss-Catalan paths having a distribution given by the Fuss-Narayana number. We prove the results using the Cycle Lemma and provide bijections among them. We also show that relative peaks are independent of the base path. In particular, valleys and high peaks can be obtained from relative peaks by fixing the base path in certain ways.