The Dyck pattern poset

TL;DR

This paper introduces the Dyck pattern poset, analyzing pattern containment in Dyck paths, providing formulas for cover relations, enumerating pattern-avoiding classes, and proposing conjectures and open problems.

Contribution

It defines the Dyck pattern poset, derives formulas for cover relations, and explores pattern-avoidance enumeration and conjectures in Dyck paths.

Findings

Formulas for the number of Dyck paths covered by or covering a given path

Enumeration of pattern-avoiding Dyck paths

Conjecture on asymptotic behavior of pattern-avoiding sequences

Abstract

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of Dyck paths covered by P, as well as for the number of Dyck paths covering P. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.