منابع مشابه
On Generalized Dyck Paths
We generalize the elegant bijective proof of the Chung Feller theorem from the paper of Young-Ming Chen [The Chung-Feller theorem revisited, Disc. Math. 308 (2008), 1328–1329].
متن کاملCounting Generalized Dyck Paths
The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0, 0) to (n, n) which is below the diagonal line y = x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0, 0) to (m, n) ∈ N2 which is below the diagonal line y...
متن کاملReturns and Hills on Generalized Dyck Paths
In 2009, Shapiro posed the following question: “What is the asymptotic proportion of Dyck paths having an even number of hills?” In this paper, we answer Shapiro’s question, as well as a generalization of the question to ternary paths. We find that the probability that a randomly chosen ternary path has an even number of hills approaches 125/169 as the length of the path approaches infinity. Ou...
متن کاملCounting Humps and Peaks in Generalized Dyck Paths
Let us call a lattice path in Z × Z from (0, 0) to (n, 0) using U = (1, k), D = (1,−1), and H = (a, 0) steps and never going below the x-axis a (k, a)-path (of order n). A super (k, a)-path is a (k, a)-path which is permitted to go below the x-axis. We relate the total number of humps in all of the (k, a)paths of order n to the number of super (k, a)-paths, where a hump is defined to be a seque...
متن کاملPattern - avoiding Dyck paths †
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 ...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2011
ISSN: 1077-8926
DOI: 10.37236/527