Dyck path Combinatorics

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Addis Ababa University


This project is all about combinatorial enumeration of Dyck paths and generalization which is intimately enumerated by Catalan numbers, Fine numbers, Motzkin numbers, bijection, and generating function. The project also gives a unified presentation and history of the previous results in the literature that mention Dyck paths. Among the settings that mention this project, section one looks about Historical background of Dyck path. In section two it is intended to count the number of Dyck paths according to several parameters, such as number of length, number of peaks, number of valleys, number of double rises, number of returns to the x-axis, the odd length of the downward path ending on the x-axis, and also number of Dyck paths that have no peaks at height two, number of Dyck paths whose first down step is followed by another down step, number of Dyck paths that have n-1 up steps, number of Dyck paths that have n-1 peaks and containing neither long slopes nor lonely peaks, number of Dyck paths with no hills, number of Dyck paths whose initial peak is at height 𝑘𝑘, whose first peak has even height and whose first peak has odd height etc. And finally the last section, presents the Dyck path statistics using both Ordinary and Bivariate generating functions



Dyck path Combinatorics