Porazdelitve : diplomsko delo

Likozar, Barbara (author), Šparl, Primož (mentor)

Porazdelitve sodijo na področje kombinatorike, v osnovi pa gre za štetje načinov, na katere lahko določen nabor označenih ali neoznačenih predmetov razporedimo v določen nabor označenih ali neoznačenih predalčkov, pri čemer so predalčki lahko prazni ali pa tudi ne. V diplomskem delu so predstavljene vse vrste porazdelitev, večji poudarek pa je na porazdelitvah, kjer ne razlikujemo ne predmetov ne predalčkov. Namen pričuječega diplomskega dela je razumljivo in logično predstaviti vrste porazdelitev in izračun njihovega števila. Števila porazdelitev neoznačenih predmetov v neoznačene predalčke so v delu bolj natančno opredeljena, predstavljenih je več načinov njihovega izračuna, kakor tudi korespondenca med obravnavanimi porazdelitvenimi števili in Youngovimi diagrami. Za števila porazdelitev neoznačenih predmetov v neoznačene neprazne predalčke je izračunana eksplicitna formula za primere, ko imamo na voljo največ tri predalčke. Predstavljeni so tudi tabelarni način zapisovanja teh števil in zanimivosti, ki smo jih odkrili med opazovanjem tabele. S pomočjo opazovanja in študija porazdelitvenih števil je predstavljenih in dokazanih tudi nekaj porazdelitvenih identitet, s katerimi prikažemo, kako iz porazdelitev enega tipa dobimo porazdelitve drugega tipa in obratno. Dokazi so večinoma izpeljani s pomočjo konjugiranih parov porazdelitev.

• Language: Slovenian
• Keywords: kombinatorika
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [B. Likozar]
• Year: 2015
• Number of pages: 37 str.
• PID: 20.500.12556/RUL-71873
• UDC: 511(043.2)
• COBISS.SI-ID: 10640713
• Publication date in RUL: 29.07.2015

LIKOZAR, Barbara, 2015, Porazdelitve : diplomsko delo [online]. Bachelor’s thesis. B. Likozar. [Accessed 22 June 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=71873

Secondary language

• Language: English
• Title: Partitions

Abstract

Partitions and partition numbers belong to the mathematical field of combinatorics. Partition numbers basically represent the number of ways in which we can arrange a specific set of marked or unmarked items into a sets of marked or unmarked boxes where the boxes can be either empty or not. This thesis contains presentations of all types of partition numbers but the focus is on the partition numbers where neither items nor boxes are distinguishable. The purpose of this thesis is to present the various types of partitions and the calculation of their number in an understandable way. Partitions of unmarked items into unmarked boxes are studied in more detail and several ways of calculating their number are presented. The thesis also explains the correspondence between partitions and the so called Young diagrams. An explicit closed formula for the number of partitions of unmarked items into unmarked non-empty boxes for cases where the number of boxes is less than or equal to three is obtained. A presentation of partition numbers in a matrix form is given and several interesting observations, that can be made from the corresponding matrix, are presented and proved. Some so called partition identities are presented and proved. They establish a correspondence between partitions of two different types. They are mainly proved using the notation of so called conjugate partitions. . Ključne besede: diskretna matematika, porazdelitve, teorija števil, kombinatorika, porazdelitvena števila, porazdelitvene identitete.

• Keywords: discrete mathematics