The Dyck pattern poset

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 Dyc...

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Published inDiscrete mathematics Vol. 321; pp. 12 - 23
Main Authors Bacher, Axel, Bernini, Antonio, Ferrari, Luca, Gunby, Benjamin, Pinzani, Renzo, West, Julian
Format Journal Article
LanguageEnglish
Published Elsevier B.V 28.04.2014
Elsevier
Subjects
Asymptotic
Asymptotic properties
Combinatorics
Counting
Covering
Dyck path
Exact enumeration
Mathematical analysis
Mathematical models
Mathematics
Pattern
Polypropylenes
Poset
Recursive
Set theory
Poset
Pattern
Exact enumeration
Dyck path
Asymptotic
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Summary: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. We also compute the generating function of Dyck paths avoiding any single pattern in a recursive fashion, from which we deduce the exact enumeration of such a class of paths. Finally, we describe the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern, we prove that the Dyck pattern poset is a well-ordering and we propose a list of open problems.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2013.12.011