Master
2021/2022
Introduction to Enumerative Combinatorics
Type:
Elective course (Joint Master's Programme with the Centre of Teaching Excellence)
Area of studies:
Mathematics
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1 year, 4 module
Mode of studies:
distance learning
Online hours:
113
Open to:
students of one campus
Instructors:
Vladimir Sharich
Master’s programme:
Joint Master's Programme with the Centre of Teaching Excellence
Language:
English
ECTS credits:
6
Contact hours:
2
Course Syllabus
Abstract
Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers
Learning Objectives
- Acquaintance with the basic notions, methods, and problems of Enumerative Combinatorics.
- Acquiring an idea of the role of Enumerative Combinatorics in other areas of mathematics (algebra, geometry, analysis, etc.)
- Acquiring the skills of applying methods and constructions of Enumerative Combinatorics to scientific research in various areas of mathematics.
- Acquiring the ability for independent study of topical mathematical literature
Expected Learning Outcomes
- Knowledge of the basic notions, methods and problems of Enumerative Combinatorics. Skills of applying methods and construction of Enumerative Combinatorics in other areas of mathematics. Experience in independent study of topical mathematical literature
Course Contents
- Permutations and binomial coefficients
- Binomial coefficients, continued. Inclusion and exclusion formula
- Linear recurrences. The Fibonacci sequence
- A nonlinear recurrence: many faces of Catalan numbers
- Generating functions: a unified approach to combinatorial problems. Solving linear recurrences
- Generating functions, continued. Generating function of the Catalan sequence Partitions.
- Euler’s generating function for partitions and pentagonal formula
- Gaussian binomial coefficients. “Quantum” versions of combinatorial identities
Bibliography
Recommended Core Bibliography
- Richard P. Stanley. (2013). Topics in algebraic combinatorics. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.21998FFA
Recommended Additional Bibliography
- Anders Björner, Kungl Tekniska, & Richard P. Stanley. (2010). A Combinatorial Miscellany by. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.3199213D