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arXiv · 2108.08299

Restricted Dyck Paths on Valleys Sequence

Rigoberto Flórez·Toufik Mansour·José L. Ramírez·Fabio A. Velandia·Diego Villamizar
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Abstract

In this paper we study a subfamily of a classic lattice path, the \emph{Dyck paths}, called \emph{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 \emph{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 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.

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Rigoberto Flórez, Toufik Mansour, José L. Ramírez, Fabio A. Velandia, Diego Villamizar. 2021-08-17. Restricted Dyck Paths on Valleys Sequence. https://arxiv.org/abs/2108.08299

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