Kombinatorni problemi distribucija

master's thesis

• Author: Barbara Polić
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-04-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu proučavaju se razne vrste distribucija koristeći model kuglica i kutija uz određene uvjete kao što je različitost kuglica, različitost kutija, ograničenost kutija itd. Za jednake kutije smo problem povezali s particijama skupa (pa time i sa Stirlingovim brojem druge vrste) u slučaju kada se kuglice razlikuju, odnosno s particijom prirodnog broja za jednake kuglice. U oba slučaja pripadni broj distribucija zadovoljava jednostavnu rekurzivnu formulu koju smo implementirali u programskom jeziku Python 3.4. Za različite kutije i različite kuglice smo koristili permutacije i kombinacije zajedno s nekim rezultatima dobivenim u slučajevima s jednakim kutijama i različitim kuglicama dok je problem s jednakim kuglicama bio složeniji te su bile potrebne FUI i FI formule. Dobivena je veza s brojem injekcija, surjekcija i funkcija a tako i formule za Stirlingov broj druge vrste i Bellov broj. Na kraju, dobiveni rezultati su bili tablično prikazani.

Abstract (english)

This thesis studies different distributions using the box-ball model with certain criteria such as inequal balls, inequal boxes, boxes with limit etc. For equal boxes we linked the problem to set partitions (Stirling number of the second kind) when the balls are inequal and respectively with partition of a natural number when the balls are equal. In both cases the number of distributions satisfies the simple recursive formula that we implemented in the programming language Python 3.4. For inequal boxes and inequal balls we used permutations and combinations together with some results from cases with equal boxes and inequal balls, while the problem with equal balls was more complex and inclusion-exclusion and generating function formulas were needed. The result is a link with the number of injections, surjections and functions, and also the formula for Stirling number of the second kind and Bell’s number. In the end, the results are shown in a table.

• Keywords: distribucije; particije skupa; Stirlingov broj; Python; Bellov broj
• Keywords (english): distributions; set partitions; Stirling number; Python; Bell’s number
• Language: croatian
• URN:NBN: urn:nbn:hr:217:224246
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-04 09:09:49