Kombinatorne primjene i asimptotski razvoji Catalanovih brojeva

undergraduate thesis

• Title (english): Combinatorial Applications and Asymptotic Expansions of Catalan numbers
• Author: Mato Manović
• Mentor: Tomislav Burić (mentor)
• Committee member: Tomislav Burić (predsjednik povjerenstva)
• Committee member: Lana Horvat Dmitrović (član povjerenstva)
• Committee member: Lenka Mihoković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Electrical Engineering and Computing; Zagreb
• Defense date and country: 2017-07-10, Croatia
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Computing

Abstract

Cilj prvog dijela ovog rada je bio pronaći i programski implementirati učestale kombinatorne probleme koji u svojoj osnovi imaju veze s Catalanovim brojevima.Neki od primjera takvih problema su putovi u cjelobrojnoj mreži, definiranje redoslijeda izvođenja operacija u algebarskom izrazu, permutacije uporabom stoga i mnogi drugi.Drugi dio rada se odnosi na dio numeričke matematike koja se bavi aproksimacijom funkcija korištenjem asimptotskog razvoja.U ovom radu za aproksimaciju Catalanovih brojeva korišten je Burić-Elezović asimptotski razvoj kvocijenta dviju gama funkcija budući da same Catalanove brojeve možemo izraziti preko kvocijenta dviju gama funkcija.Izračun koeficijenata asimptotskog razvoja Catalanovih brojeva bilo je potrebno implementirati u prikladnom matematičkom programskom alatu.

Abstract (english)

The goal of the first part of thesis was to find and programmatically implement frequent combinatorial problems whose basis is related with Catalan numbers. Some problem examples are lattice paths along the edges of the grid, defining the sequence of operations in algebraic expression and permutations using stack. The second part of thesis is about the segment of numerical mathematics used for approximation of functions by using asymptotic expansion. In this thesis, the Burić-Elezović asymptotic expansion of the quotioent of two gamma functions was used to approximate Catalan numbers as they can be expressed over the quotient of two gamma functions. Calculating the asymptotic expansion coefficients of Catalan numbers required implementation in suitable mathematical program tool.

• Keywords: Catalanovi brojevi; asimptotski razvoj; Burić-Elezović teorem; Bernoullijevi polinomi; gama funkcija; kombinatorni problemi; funkcija izvodnica
• Keywords (english): Catalan numbers; asymptotic expansion; Burić-Elezović theorem; Bernoulli polynomials; gamma function; combinatorial problems; generating function
• Language: croatian
• URN:NBN: urn:nbn:hr:168:861623
• Study programme: Title: Computing; Study programme type: university; Study level: undergraduate; Academic / professional title: sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka računarstva (sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka računarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Closed access
• Created on: 2019-03-13 22:36:58