Updated Problem Set 6 is available here.

**class meets:** Tuesday and Thursday, 2:30 - 4 pm,
room 2-136

**instructor:**
Alexander Postnikov

**office hour:** Tuesday 4 - 5 pm

**description:**

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

**topics:**

pigeon-hole principle,
mathematical induction,
permutations,
binomial theorem,
compositions,
partitions,
Stirling numbers,
inclusion-exclusion principle,
recurrence relations,
generating functions,
Catalan numbers,
graphs, trees, Eulerian walks, Hamiltionian cycles,
matrix-tree theorem,
electrical networks,
graph colorings, chromatic polynomials,
(and if time allows)
Polya counting, Ramsey theory,
pattern avoidance, probabilistic method,
partial orders, combinatorial algorithms ...

**course level:** undergraduate

**recommended textbook:**

**-**
*A Walk through Combinatorics*,
1st edition
or 2nd edition,
by Miklos Bona,
World Scientific.

The students will not be required to buy this textbook.

**grading:**
Problem sets due every two weeks 50% +
3 inclass quizes 50%.
There will be no final exam.

**problem sets:**

- Problem Set 1 (due 09/20/2007)
- Problem Set 2 (due 10/04/2007) answers
- Problem Set 3 (due 10/23/2007) answers
- Problem Set 4 (due 11/01/2007) answers
- Problem Set 5 (due 11/20/2007) answers
- Problem Set 6 (due 12/11/2007)

**practice quizes:**

- Quiz 1: quiz A and quiz B
- Quiz 2: quiz A, quiz B, and quiz C with answers
- Quiz 3: quiz A, quiz B, and quiz C with answers

**syllabus:** (tentative)

- R 09/06/2007. Pigeonhole principle. Chapter 1.
- T 09/11/2007. Mathematical induction. Chapter 2.
- R 09/13/2007. Permutations. Rook placements.
Binomial coefficients. Lattice paths.
Chapter 3.
- T 09/18/2007. Binomial theorem. Chapter 4.
- R 09/20/2007. Compositions. Integer partitions.
Sections 5.1 and 5.3.
Problem Set 1 is due.
- T 09/25/2007. Set partitions. Section 5.2.
- R 09/27/2007. Cycle structure of permutations. Chapter 6.
- T 10/02/2007. Quiz 1.
- R 10/04/2007. Inclusion-exclusion principle. Chapter 7.
Problem Set 2 is due.
T 10/09/2007. no classes (Columbus day)

- R 10/11/2007. Inclusion-exclusion (cont'd).
- T 10/16/2007. Generating functions and recurrence relations.
Chapter 8.
- R 10/18/2007. Generating functions (cont'd). Ordinary generating
functions.
Problem Set 3 is due.
- T 10/23/2007. Generating functions (cont'd). Exponential
generating functions.
- R 10/25/2007. Catalan numbers.
- T 10/30/2007. Catalan miscellany.
- R 11/01/2007. Quiz 2.
Problem Set 4 is due.
- T 11/06/2007. Graphs. Eulerian walks. Hamiltionian cycles.
Chapter 9.
- R 11/08/2007. Trees. Counting trees. Chapter 10.
- T 11/13/2007. Minimum-weight trees. Greedy algorithm.
Section 10.2.
- R 11/15/2007. Matrix-tree theorem. Section 10.3.
- T 11/20/2007. Electrical networks.
Problem Set 5 is due.
R 11/22/2007. no classes (Thanksgiving)

- T 11/27/2007. Eulerian digraphs.
- R 11/29/2007.
Graph colorings. Chromatic polynomials.
Matchings.
Chapter 11.
- T 12/04/2007. Quiz 3.
- R 12/06/2007. Polya counting. Ramsey theory.
- T 12/11/2007. Pattern avoidance. Probabilistic method. Problem Set 6 is due.

**additional information:**