Particije prirodnih brojeva

master's thesis

• Author: Martina Barić
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Particija prirodnog broja \(n\) je svaka uređena \(k\)-torka prirodnih brojeva \((a_1,a_2,\ldots,a_k)\) za koju je \[a_1\geq a_2\geq\cdots\geq a_k\] i \[ n=a_1+a_2+\cdots+a_k.\] Broj svih particija prirodnog broja \(n\) označava se s \(p(n)\). Tako je \(p(4)=5\), budući da \(4\) možemo prikazati kao sljedeće sume \(4\), \(3 + 1\), \(2 + 2\), \(2 + 1 + 1\), \(1 + 1 + 1 + 1\). Iako su particije izuzetno jednostavan matematički pojam, one su bitan matematički objekt koji zadovoljava niz zanimljivih i neočekivanih svojstava. U ovom radu će se opisati kako se particije mogu vizualizirati pomoću Ferrerovih dijagrama. Također, iskazati će se i dokazati neki identiteti za \(p(n)\), na primjer Eulerov identitet, koristeći metodu bijekcije.

Abstract (english)

A partition of a positive integer \(n\) is a \(k\)-tuple \((a_1,a_2,\ldots,a_k)\) consisting of positive integers such that \[a_1\geq a_2\geq\cdots\geq a_k\] and \[ n=a_1+a_2+\cdots+a_k.\] The number of partitions of \(n\) is given by the partition function \(p(n)\). For instance, \(p(4)=5\) since \(4\), \(3 + 1\), \(2 + 2\), \(2 + 1 + 1\) and \(1 + 1 + 1 + 1\) represent all partitions of \(4\). Although partitions seems to be very simple mathematical concept, they are important mathematical objects that have many interesting and unexpected properties. In this thesis, we describe how integer partitions can be graphically visualized with Ferrers graphs. Also, we give and prove some partition identities, like Euler's identity, by using the bijective method.

• Keywords: prirodni broj. Ferrerovi dijagrami; Eulerov identitet
• Keywords (english): positive integer; Ferrers graphs; Euler's identity
• Language: croatian
• URN:NBN: urn:nbn:hr:217:476747
• 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: 2017-10-30 13:25:00