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

TL;DR

This paper explores the relationships between restricted Dyck paths, non-crossing partitions, and polyominoes, providing generating functions to count various combinatorial structures and revealing new connections among them.

Contribution

It introduces new links between restricted Dyck paths, non-crossing partitions, and polyominoes, along with explicit generating functions for their enumeration.

Findings

Established connections between restricted Dyck paths and non-crossing partitions.

Derived generating functions for counting these combinatorial objects.

Identified relationships with specific subclasses of polyominoes.

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.