کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4647369 | 1632421 | 2014 | 12 صفحه PDF | دانلود رایگان |
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.
Journal: Discrete Mathematics - Volume 321, 28 April 2014, Pages 12–23