18.314   M.I.T.   Fall 2005

## [[[c][oM][bin]][[At][OR]]][[i][AL]] .a.n.a.l.y.s.i.s.

Class meets: Tuesday and Thursday, 1-2:30 pm, room 12-102

Instructor: Alexander Postnikov

Office hour: Tuesday 3-4 pm

Description:
From Catalog: Combinatorial problems and methods for their solution. Prior experience with abstraction and proofs helpful. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Network algorithms, extremal combinatorics.

Textbook:
*  A Walk through Combinatorics, Miklos Bona, World Scientific, 2002.
Author's errata, errata by R. Ehrenborg, errata by R. Stanley.

*  Enumerative Combinatorics, Vol 1 and Vol 2, by R. P. Stanley, Cambridge University Press, 1996 and 1999.
*  Introductory Combinatorics, R. Brualdi, 3rd or 4th edition, Prentice Hall.

Grading: 3 inclass exams 50% total, problem sets (due every second Tuesday) 50%, + bonuses.

Problem Sets: Due every second Tuesday

Practice Exams:

Bonuses: You can get a grading bonus if you write an article in combinatorics, invent a new interesting integer sequence and publish it in Sloan's Encyclopedia of Integer Sequences, find a new interpretation of the Catalan numbers that is not listed in Stanley's EC2 and Catalan Addendum, or solve some of the bonus problems in problem sets.

Syllabus: (tentative)

1. R 09/08/2005. Introduction. Pigeonhole principle. Chapter 1.

2. T 09/13/2005. Mathematical induction. Chapter 2.

3. R 09/15/2005. Permutations. Chapter 3.

4. T 09/20/2005. Binomial theorem. Chapter 4. Problem Set 1 is due.

5. R 09/22/2005. Compositions. Integer Partitions. Chapter 5.

6. T 09/27/2005. Set partitions.

7. R 09/29/2005. Cycles in permutations. Stirling numbers. Chapter 6.

8. T 10/04/2005. Exam 1. Problem Set 2 is due.

9. R 10/06/2005. Inclusion-exclusion principle. Chapter 7.

T 10/11/2005. no classes (Columbus day)

10. R 10/13/2005. Inclusion-exclusion (cont'd). Mobius inversion.

11. T 10/18/2005. Recurrence relations.

12. R 10/20/2005. Generating functions. Chapter 8.

13. T 10/25/2005. Generating functions (cont'd). Problem Set 3 is due.

14. R 10/27/2005. Catalan numbers.

15. T 11/01/2005. Generating functions (cont'd). Problem Set 4 is due.

16. R 11/03/2005. Exam 2.

17. T 11/08/2005. Graphs. Eulerian walks. Hamiltionian cycles. Chapter 9.

18. R 11/10/2005. Trees. Counting trees. Chapter 10.

19. T 11/15/2005. Matrix-tree theorem.

20. R 11/17/2005. Matrix-tree theorem (cont'd). Problem Set 5 is due.

21. T 11/22/2005. Guest lecture by Igor Pak

R 11/24/2005. no classes (Thanksgiving)

22. T 11/29/2005. Matrix-tree theorem and Electrical networks.

23. R 12/01/2005. Electrical networks (cont'd). Eulerian digraphs and BEST theorem.

24. T 12/06/2005. Graph colorings. Bipartite graphs and matchings. Chromatic polynomials. Chapter 11.

25. R 12/08/2005. Exam 3. Problem Set 6 is due.

26. T 12/13/2005. ... Polya counting. Ramsey theory. Probabilistic method.