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arXiv · 2308.02059

Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions

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

A \emph{Dyck path} is a lattice path in the first quadrant of the $xy$-plane that starts at the origin, ends on the $x$-axis, and consists of the same number of North-East steps $U$ and South-East steps $D$. A \emph{valley} is a subpath of the form $DU$. A Dyck path is called \emph{restricted $d$-Dyck} if the difference between any two consecutive valleys is at least $d$ (right-hand side minus left-hand side) or if it has at most one valley. In this paper we give some connections between restricted $d$-Dyck paths and both, the non-crossing partitions of $[n]$ and some subfamilies of polyominoes. We also give generating functions to count several aspects of these combinatorial objects.

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Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, Diego Villamizar. 2023-08-03. Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions. https://arxiv.org/abs/2308.02059

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