Eulerovski i Narayanini brojevi

master's thesis

• Author: Karlo Lončarević
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Ana Prlić (član povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Za permutaciju \(w=w_1w_2\ldots w_n\in S_n\) kažemo da je prirodni broj \(i\) pad ako je \(w_i>w_{i+1}\). Eulerovski broj označen s \(\genfrac{<}{>}{0pt}{}{n}{k}\) je broj permutacija u \(S_n\) s točno \(k\) padova. Označimo li sa \(S_n(231)\) permutacije iz \(S_n\) u kojima se ne pojavljuje uzorak 231, Narayanin broj je broj permutacija iz \(S_n(231)\) s \(k\) padova. Ovaj diplomski rad se sastoji od tri poglavlja. U prvom poglavlju navodimo neke osnovne pojmove i rezultate potrebne za razumijevanje nastavka rada. Drugo poglavlje uvodi Eulerovske brojeve koje proučavamo počevši od definicije i osnovnih svojstava. Uvode se Eulerovski polinomi te dokazuju dva važna identiteta, a poglavlje završavamo jednim geometrijskim problemom u kojemu se neočekivano pojavljuju Eulerovski brojevi. Treće poglavlje bavi se Narayaninim brojevima koji su u bliskoj vezi s Eulerovskim brojevima. U ovom poglavlju važno mjesto imaju i Catalanovi brojevi kojima su Narayanini brojevi svojevrsno profinjenje. Prikazano je nekoliko primjera i dobivena funkcija izvodnica za Narayanine brojeve.

Abstract (english)

For any permutation \(w=w_1w_2\ldots w_n\in S_n\), we call a positive integer \(i\) a descent if \(w_i>w_{i+1}\). Eulerian number denoted \(\genfrac{<}{>}{0pt}{}{n}{k}\) is the number of permutations in \(S_n\) with exactly \(k\) descents. Denoting by \(S_n(231)\) the set of permutations in \(S_n\) not containing the pattern 231, Narayana number \(N_{n,k}\) is the number of permutations in \(S_n(231)\) with \(k\) descents. This Master’s thesis consists of three chapters. The first chapter contains some introductory notions and results needed in the rest of the work. The second chapter examines Eulerian numbers starting from their definition and basic properties. We introduce the Eulerian polynomials and prove two important identities finishing this part with a problem from geometry where Eulerian numbers make an unexpected appearance. The third chapter treats the Narayana numbers which are closely connected to Eulerian numbers. Catalan numbers also play an important role in this chapter since Narayana numbers are shown to be their refinement. Several examples where Narayana numbers appear are presented and their generating function is obtained.

• Keywords: Eulerovski brojevi; Narayanini brojevi; Eulerovski polinomi; Catalanovi brojevi
• Keywords (english): Eulerian numbers; Narayana numbers; Eulerian polynomials; Catalan numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:749056
• 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: 2021-03-03 11:14:07