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Introduction to Enumerative Combinatorics

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
3
Кредиты
Статус:
Курс по выбору
Когда читается:
2-й курс, 4 модуль

Преподаватель

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

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

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

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
Assessment Elements

Assessment Elements

  • non-blocking 8 quizes
  • non-blocking final exam
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    The final grade is the summary of average for 8 quizes (50%) and the final exam (50%).
Bibliography

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