Pattern-avoiding Dyck paths

We introduce the notion of $\textit{pattern}$ 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 $\textit{Dyck patter...

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Published inDiscrete Mathematics and Theoretical Computer Science Vol. DMTCS Proceedings vol. AS,...; no. Proceedings; pp. 683 - 694
Main Authors Bernini, Antonio, Ferrari, Luca, Pinzani, Renzo, West, Julian
Format Journal Article Conference Proceeding
LanguageEnglish
Published DMTCS 01.01.2013
Discrete Mathematics and Theoretical Computer Science
Discrete Mathematics & Theoretical Computer Science
SeriesDMTCS Proceedings
Subjects
[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
Computer Science
Discrete Mathematics
dyck path
enumeration
pattern containment relation
enumeration
pattern containment relation
Dyck path
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Summary:We introduce the notion of $\textit{pattern}$ 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 $\textit{Dyck pattern poset}$. 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.
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.2334