Eulerovi brojevi

master's thesis

• Title (english): Eulerian Numbers
• Author: Filip Vinković
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Mea Bombardelli (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2025-02-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics; Discrete and Combinatorial Mathematics

Abstract

Za prirodni broj \(i\) kažemo da je pad u permutaciji \(p=p_{1}p_{2}\ldots p_{n}\in S_{n}\) ako vrijedi \(p_{i}>p_{i+1}.\) Eulerov broj \(\left\langle n\atop k \right\rangle\) označava broj permutacija stupnja \(n\) s točko \(k\) padova. U radu prvo objašnjavamo matematičke principe poput prebrojavanja, permutacija i binomnih koeficijenata kako bismo osigurali podlogu za razumijevanje nastavka rada. Eulerove brojeve definiramo kroz koncept padova u permutacijama, što omogućuje istraživanje njihovih osnovnih svojstava i različitih identiteta koje zadovoljavaju. Posebno su naglašene rekurzivne metode za računanje Eulerovih brojeva, kao i eksplicitne formule. Istražena je i veza između Eulerovih i Stirlingovih brojeva druge vrste, gdje su navedene formule za njihov međusobni odnos. Nakon toga su definirane funkcije izvodnice čiji su članovi Eulerovi brojevi, a u posebnom se dijelu analizira niz Eulerovih brojeva te dokazuje svojstvo log-konkavnosti. Na kraju rada uvodimo koncept Eulerovih brojeva druge vrste i iskazujemo neka njihova osnovna svojstva, kao i eksplicitnu formulu za njihovo računanje.

Abstract (english)

We say that a positive integer \(i\) is a descent of the permutation \(p=p_{1}p_{2}\ldots p_{n}\in S_{n}\) if \(p_{i}>p_{i+1}\) holds. Eulerian number \(\left\langle n\atop k \right\rangle\) denotes the number of \(n\)-permutations with exactly \(k\) descents. The paper begins by explaining basic mathematical principles such as counting, permutations, and binomial coefficients, providing the groundwork for understanding the remainder of the study. Eulerian numbers are defined through the concept of descents in permutations, enabling the exploration of their fundamental properties and various identities they satisfy. Recursive methods for calculating Eulerian numbers, as well as explicit formulas, are emphasized. The relationship between Eulerian numbers and Stirling numbers of the second kind is also examined, with formulas provided for their interconnection. Generating functions whose terms are Eulerian numbers are then defined, and log-concavity of Eulerian numbers is proved. Finally, the concept of second order Eulerian numbers is introduced, outlining their basic properties and presenting an explicit formula for their computation.

• Keywords: Eulerovi brojevi; kombinatorika; Euler
• Keywords (english): Eulerian numbers; combinatorics; Euler
• Language: croatian
• URN:NBN: urn:nbn:hr:217:322327
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-02-09 16:09:14