Elementary Combinatorics


Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designsand matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in analgebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry,[1] and combinatorics also has many applications in mathematical optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

Reading List:

1- A Walk Through Combinatorics; An Introduction to Enumeration and Graph Theory, Third Edition (Miklos Bona)(2011)

2- Introductory Combinatorics (Richard A. Brualdi)(2009)

Course Contents:

1- The Pigeonhole Principle 

2-The Method of Mathematical Induction 

3-Generating Permutations and combinations

4-The Inclusion-Exclusion Principle and Applications 

5-Recurrence Relations and Generating Functions

6-Graph theory

7-Ramsey theory

8-Combinatorial designs

9-Polya counting

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Grading Policy:

Homework (%10)

First Midterm Exam (%40)

Final Exam (%50)

Course Notes and Other Materials:

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