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

**Instructor:**
Alexander Postnikov

**Office hour:** Tuesday 3-4 pm

**Grader:** Fu Liu

**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.

**Course Level:** Undergraduate

**Textbook:**

*****
*A Walk
through Combinatorics,* Miklos Bona,
World Scientific, 2002.

Author's errata,
errata by R. Ehrenborg,
errata by R. Stanley.

**Additional Reading:**

*****
*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

- Problem Set 1 (due 09/22/2005)
- Problem Set 2 (due 10/04/2005)
- Problem Set 3 (due 10/25/2005)
- Problem Set 4 (due 11/01/2005)
- Problem Set 5 (due 11/17/2005)
- Problem Set 6 (due 12/08/2005)

**Practice Exams:**

- Exam 2: Practice Exam and Another Practice Exam.
- Exam 3: Practice Exam and Another Practice Exam.

**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)

- R 09/08/2005. Introduction. Pigeonhole principle. Chapter 1.
- T 09/13/2005. Mathematical induction. Chapter 2.
- R 09/15/2005. Permutations. Chapter 3.
- T 09/20/2005. Binomial theorem. Chapter 4. Problem Set 1 is due.
- R 09/22/2005. Compositions. Integer Partitions. Chapter 5.
- T 09/27/2005. Set partitions.
- R 09/29/2005. Cycles in permutations. Stirling numbers. Chapter 6.
- T 10/04/2005. Exam 1. Problem Set 2 is due.
- R 10/06/2005. Inclusion-exclusion principle. Chapter 7.
T 10/11/2005. no classes (Columbus day)

- R 10/13/2005. Inclusion-exclusion (cont'd). Mobius inversion.
- T 10/18/2005. Recurrence relations.
- R 10/20/2005. Generating functions. Chapter 8.
- T 10/25/2005. Generating functions (cont'd). Problem Set 3 is due.
- R 10/27/2005. Catalan numbers.
- T 11/01/2005. Generating functions (cont'd). Problem Set 4 is due.
- R 11/03/2005. Exam 2.
- T 11/08/2005. Graphs. Eulerian walks. Hamiltionian cycles.
Chapter 9.
- R 11/10/2005. Trees. Counting trees. Chapter 10.
- T 11/15/2005. Matrix-tree theorem.
- R 11/17/2005. Matrix-tree theorem (cont'd). Problem Set 5 is due.
- T 11/22/2005. Guest lecture by Igor Pak
R 11/24/2005. no classes (Thanksgiving)

- T 11/29/2005. Matrix-tree theorem and Electrical networks.
- R 12/01/2005. Electrical networks (cont'd).
Eulerian digraphs and BEST theorem.
- T 12/06/2005. Graph colorings. Bipartite graphs and matchings.
Chromatic polynomials.
Chapter 11.
- R 12/08/2005. Exam 3.
Problem Set 6 is due.
- T 12/13/2005. ... Polya counting. Ramsey theory. Probabilistic method.