Eulerian numbers and Coxeter groups : magistrsko delo

Borges Santos da Costa, João Manuel (author), Konvalinka, Matjaž (mentor)

Eulerian numbers arise ubiquitously in various areas of mathematics. Indeed, this number sequence gives us the possibility to find several combinatorial invariants that characterize a panoply of geometric and algebraic structures. We will give a walkthrough on this beautiful counting sequence, starting with basic definitions and properties that are at the core of its combinatorial structure. Then, we introduce the notion of a partially ordered set and hyperplane arrangement, giving examples where the eulerian numbers naturally arise. Finally, there is a brief introduction to the theory of Coxeter groups, and, most importantly, how we can characterize them by using Eulerian numbers.

• Language: English
• Keywords: Eulerian numbers, Eulerian polynomials, partial order, hyperplane arrangements, Coxeter groups
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-145213
• UDC: 519.1
• COBISS.SI-ID: 148419587
• Publication date in RUL: 13.04.2023

Secondary language

• Language: Slovenian
• Title: Eulerska števila in Coxeterjeve grupe

Abstract

Eulerska števila se pojavljajo povsod na različnih področjih matematike. Dejansko nam to številsko zaporedje daje možnost, da najdemo več kombinatoričnih invariant, ki karakterizirajo več geometrijskih in algebraičnih struktur. Predstavili bomo to čudovito preštevalno zaporedje, začenši z osnovnimi definicijami in lastnostmi, ki so jedro njegove kombinatorične strukture. Nato uvedemo pojem delno urejene množice in razporeditve hiperravnine ter navedemo primere, kjer se eulerska števila naravno pojavijo. Na koncu je kratek uvod v teorijo Coxeterjevih skupin in, kar je najpomembneje, kako jih lahko karakteriziramo z uporabo eulerska števil.

• Keywords: eulerska števila, eulerski polinomi, delna urejenost, razporeditve hiperravnin, Coxeterjeve grupe