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The Dyck pattern poset

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 PP, we determine a formula for the number of Dyck paths covered by PP, as well as for the number of Dyck paths covering PP. 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.
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https://hal.archives-ouvertes.fr/hal-00989619
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Submitted on : Monday, May 12, 2014 - 3:00:22 PM
Last modification on : Wednesday, December 8, 2021 - 3:54:35 AM
Long-term archiving on: : Tuesday, August 12, 2014 - 11:25:40 AM

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Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani, et al.. The Dyck pattern poset. Discrete Mathematics, Elsevier, 2014, 321, pp.Pages 12-23. ⟨10.1016/j.disc.2013.12.011⟩. ⟨hal-00989619⟩

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