18.314   M.I.T.   Fall 2004

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

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.


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:

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.