Catalan numbers

Catalanovi brojevi javljaju se pri prebrojavanju iznenađujuće mnogo kombinatornih i drugih matematičkih objekata. U ovom radu detaljnije su opisani problem triangulacije konveksnog n-terokuta, problem zagrada, problem binarnih stabala, problem putova u cjelobrojnoj mreži, problem Dyckovih planinskih putova te Bertrandov problem glasovanja. Naposlijetku je pokazano da rješenja tih problema odgovaraju n-tom Catalanovom broju. Radi lakšeg manipuliranja nizom brojeva, izvedena je i odgovarajuća funkcija izvodnica te je tako dobivena formula za računanje općeg člana niza. Konačno, navedene su još neke zanimljive interpretacije Catalanovih brojeva.Catalan’s numbers appear in various combinatorical and other mathematical objects. In this paper convex polygon triangulation as well as balanced parentheses, problem of binary trees, problem of lattice paths, problem of Dyck paths and Bertrand’s ballot problem are being analysed. Ultimately, it was shown that the solution to each problem corresponds to the \(n^{th}\) Catalan number. In order to manipulate these numbers in an easier way, there is a generating function which is used for calculating the general number of the sequence. Finally, there are shown some other interesting interpretations of Catalan’s numbers