Journal of Integer Sequences, Vol. 21 (2018), Article 18.9.7

On Enumeration of Dyck Paths with Colored Hills

Milan Janjić
Department of Mathematics and Informatics
University of Banja Luka
Banja Luka, 78000
Republic of Srpska, BA


We continue to investigate the properties of the earlier defined functions fm and gm, which depend on an initial arithmetic function f0. In this paper, values of f0 are the Fine numbers. We investigate functions fi, gi, (i = 1, 2, 3, 4), and show that these functions count Dyck paths having hills in different colors. For each function, we also derive an explicit formula. We also prove several results which mutually connect these functions. It appears that g2 and g3 are well-known objects called the Catalan triangles.

We finish with two identities relating different kind of combinatorial objects.

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(Concerned with sequences A000108 A000957 A001700 A009766 A033184 A035324 A039598 A049027 A065600 A065601 A294527.)

Received May 29 2018; revised version received October 18 2018. Published in Journal of Integer Sequences, December 13 2018.

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