• 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.

• 대한수학회지
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• 제34권1호
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• pp.181-193
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• 1997

The enumeration of various classes of paths in the real plane has an important implications in the area of combinatorics wit statistical applications. In 1887, D. Andre [3, pp. 21] first solved the famous ballot problem, formulated by Berttand [2], by using the well-known reflection principle which contributed tremendously to resolve the problems of enumeration of various classes of lattice paths in the plane. First, it is necessary to state the definition of NSEW-paths in the palne which will be employed throughout the paper. From [3, 10, 11], we can find results concerning many of the basics discussed in section 1 and 2.

• 대한수학회보
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• 제51권4호
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• pp.1229-1240
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• 2014

In this paper we provide three results involving large Schr$\ddot{o}$der paths. First, we enumerate the number of large Schr$\ddot{o}$der paths by type. Second, we prove that these numbers are the coefficients of a certain symmetric function defined on the staircase skew shape when expanded in elementary symmetric functions. Finally we define a symmetric function on a Fuss path associated with its low valleys and prove that when expanded in elementary symmetric functions the indices are running over the types of all Schr$\ddot{o}$der paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.