Séminaire Lotharingien de Combinatoire, 87B.6 (2023), 20 pp.

Rigoberto Flórez, Toufik Mansour, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar

Restricted Dyck paths on valleys sequence

Abstract. In this paper we study a subfamily of a classic lattice path, the Dyck paths, called restricted d-Dyck paths, in short d-Dyck. A valley of a Dyck path P is a local minimum of P; if the difference between the heights of two consecutive valleys (from left to right) is at least d, we say that P is a restricted d-Dyck path. The area of a Dyck path is the sum of the absolute values of y-components of all points in the path. We find the number of peaks and the area of all paths of a given length in the set of d-Dyck paths. We give a bivariate generating function to count the number of the d-Dyck paths with respect to the semi-length and number of peaks. After that, we analyze in detail the case d=-1. Among other things, we give both the generating function and a recursive relation for the total area.

Received: August 20, 2021. Revised: August 3, 2023. Published: August 31, 2023.

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If you need to cite this article, here is a bibtex:

@article{FlMaRaVeVi23,
author = {Fl{\'o}rez, Rigoberto and Mansour, Toufik and Ram{\'{\i}}rez, Jos{\'e} L. and Velandia, Fabio A. and Villamizar, Diego},
title = {{Restricted Dyck paths on valleys sequence}},
fjournal = {S{\'e}minaire Lotharingien de Combinatoire},
journal = {S{\'e}min. Lothar. Combin.},
issn={1286-4889},
volume = {87B},
number= {6},
pages = {1--20},
year = {2023},
language = {English},
doi={10.48550/arXiv.2108.08299},
}