combinatorial analysis

Class meets: Tuesday and Thursday, 1-2:30 pm, room 4-153

Instructor: Alexander Postnikov

Office hour: Thursday, 4-5 pm

Grader: Pavlo Pylyavskyy

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

Course Level: Undergraduate

Recommended textbooks:

*  Introductory Combinatorics, R. Brualdi, 3rd or 4th edition, Prentice Hall.
*  A Walk through Combinatorics, M. Bona, World Scientific, 2002.

Further Reading:
*  Enumerative Combinatorics, Vol 1 and Vol 2, by R. P. Stanley, Cambridge University Press, 1996 and 1999.

Grading: 3 quizes 25% each, problem sets 25%, + bonuses.

Problem Sets: Due every second Thursday.

• Problem Set 1 (due 09/23/2004): PostScript and PDF
• Problem Set 2 (due 10/07/2004): PostScript and PDF
• Problem Set 3 (due 10/21/2004): PostScript and PDF
• Problem Set 4 (due 11/09/2004): PostScript and PDF
• Problem Set 5 (due 12/09/2004): PostScript and PDF

Exams:

• Practice problems for Exam 1: PostScript and PDF
• Exam 1 (10/07/2004): PostScript and PDF
• Practice problems for Exam 2: PDF (problems from last year)
• Exam 2 (11/04/2004): PostScript and PDF
• Practice problems for Exam 3: PostScript and PDF

Bonuses: You can get a grading bonus if you (a) write an article in combinatorics; (b) invent a new interesting integer sequence and publish it in Sloan's Encyclopedia; (c) find a new interpretation of the Catalan numbers that is not listed in Stanley's EC2 and Catalan Addendum; (d) solve some of the optional bonus problems in problem sets; or (e) solve some of the problems from the following list of

7 open problems: PostScript and PDF

Useful links:

• Neil Sloan's on-line Encyclopedia of Integer Sequences
• Richard Stanley's EC page that includes a list of problems on Catalan numbers (PS and PDF) together with Catalan Addendum (PS and PDF)

Lectures: (tentative schedule)

1. R 09/09/2004. Course overview. Introduction.

2. T 09/14/2004. Pigeonhole principle.

3. R 09/16/2004. Mathematical induction.

4. T 09/21/2004. Permutations.

5. R 09/23/2004. Permutations (cont'd). PS1 is due.

6. T 09/28/2004. Binomial theorem.

7. R 09/30/2004. Compositions.

8. T 10/05/2004. Partitions.

9. R 10/07/2004. Quiz I. PS2 is due.

10. T 10/12/2004. Inclusion-exclusion.

11. R 10/14/2004. Inclusion-exclusion (cont'd). Mobius inversion.

12. T 10/19/2004. Recurrence relations.

13. R 10/21/2004. Generating functions. PS3 is due.

14. T 10/26/2004. Exponential generating functions.

15. R 10/28/2004. Catalan numbers.

16. T 11/02/2004. No class

17. R 11/04/2004. Quiz II.

18. T 11/09/2004. Matchings. Eulerian walks. Hamiltionian cycles. PS4 is due.

    R 11/11/2004. No class: Veterans Day

19. T 11/16/2004. Trees. Counting trees.

20. R 11/18/2004. Matrix-tree theorem.

21. T 11/23/2004. Matrix-tree theorem (cont'd).

    R 11/25/2004. No class: Thanksgiving

22. T 11/20/2004. Chromatic numbers. Planar graphs.

23. R 12/02/2004. Quiz III.

24. T 12/07/2004. Polya counting.

25. R 12/09/2004. Probabilistic method. PS5 is due.