q-analogoni kombinatoričkih brojeva i identiteta

master's thesis

• Author: Dunja Jagetić
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Saša Singer (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U matematici postoji mnogo istaknutih nizova brojeva, poput parnih i neparnih brojeva, prostih brojeva, kvadrata i kubova brojeva, Fibonaccijevih brojeva itd. U ovom radu proučavamo neke brojeve specifične za područje kombinatorike, i to binomne koeficijente, \(q\)-binomne koeficijente (Gaussove koeficijente), Stirlingove brojeve, \(q\)-Stirlingove brojeve te Bellove brojeve. Binomni koeficijenti \(\binom{n}{k}\) brojevi su koji prebrojavaju \(k\)-člane podskupove konačnog \(n\)-članog skupa. Njihovi \(q\)-analogoni, Gaussovi koeficijenti \({n \brack k}_q\) prebrojavaju \(k\)-dimenzionalne potprostore \(n\)-dimenzionalog vektorskog prostora nad konačnim poljem reda \(q\). Stirlingovi brojevi druge vrste \({n \brace k}\) prebrojavaju \(k\)-particije \(n\)-članog skupa, odnosno na koliko načina možemo podijeliti \(n\)-člani skup u \(k\) nepraznih podskupova. Bellov broj \(B_n\) je ukupan broj svih particija nekog \(n\)-članog skupa. \(q\)-Stirlingov broj \({n \brace k}_q\) definira se kao težina skupa svih \(k\)-particija nekog \(n\)-članog skupa. U diplomskom radu navodimo razne identitete i relacije koje vrijede za spomenute kombinatoričke brojeve. Naglašavamo sličnosti i razlike između brojeva koji prebrojavaju konačne skupove i njihovih analogona nad konačnim poljima.

Abstract (english)

In mathematics there are many interesting number sequences, such as odd and even numbers, prime numbers, square and cube numbers, Fibonacci numbers etc. This thesis studies certain numbers which are specific for the area of combinatorics, in particular binomial coefficients, \(q\)-binomial coefficients (Gaussian coefficients), Stirling numbers and \(q\)-Stirling numbers, along with brief mention of Bell numbers. The binomial coefficient \(\binom {n}{k}\) is the number of \(k\)-element subsets of a set of \(n\) elements. The \(q\)-analogue of the binomial coefficient, also called a Gaussian coefficient \({n \brack b}_q\), counts the number of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space over the finite field of order \(q\). The Stirling number of the second kind \({n \brace k}\) counts \(k\)-partitions of an \(n\)-element set, in other words the ways to divide a set of \(n\) elements into \(k\) nonempty subsets. Bell number \(B_n\) counts the number of all partitions of an \(n\)-element set. The \(q\)-Stirling number \({n \brace k}_q\) is defined as a weight of the set of all \(k\)-partitions of an \(n\)-element set. In the thesis we give various identities and relations valid for the aforementioned combinatorial numbers. We emphasise similarities and differences between numbers enumerating finite sets and their analogues over finite fields.

• Keywords: nizovi brojeva; kombinatorika; binomni koeficijenti; Gaussovi koeficijenti; Stirlingovi brojevi; Bellovi brojevi
• Keywords (english): number sequences; combinatorics; binomial coefficients; Gaussian coefficients; Stirling numbers; Bell numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:492625
• 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: 2019-01-31 12:58:26