- Textbook
- Problem Sets due on Wednesdays.
- Selected Solutions
- Other Handouts and Reading Assignments
- Some Reading Assignments from Past Years

- Combinatorics Seminar at UW
- Recent preprints on research in Combinatorics from the arXiv.
- Mathscinet Index to all published research in mathematics. Includes 3,206,221 total publications as of 9/30/2015 going back as far as 200 years ago.
- On-Line Encyclopedia of Integer Sequences (OEIS) Amazing collection of integer sequences started by Neil Sloane. Try to "listen" to a sequence! Or "browse" the Best Sequences.
- Generatingfunctionology by Herb Wilf
- Wikipedia Dictionary of almost all terminology. This is a great teaching and learning tool. Check out the extensive article on Combinatorics.
- SAGE: Open Source Mathematical Software A collection of mathematical tools to do symbolic computation, combinatorial algorithms, graph manipulation, exact linear algebra, etc. Plus, its being developed right here at UW!
- Maple on the Web - Links to almost every Maple related web site.
- Graphviz Package Graph drawing software. Here is my dot file for Bruhat order.

Summary:This three quarter topics course on Combinatorics includes Enumeration, Graph Theory, and aspects of Algebraic and Geometric Combinatorics. Combinatorics has connections to all areas of mathematics, industry and many other sciences including biology, physics, computer science, and chemistry. We have chosen core areas of study which should be relevant to a wide audience. The main distinction between this course and its undergraduate counterpart will be the pace and depth of coverage. In addition we will assume students have a substantial knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research. New students are welcome to join in Winter or Spring with approval from the instructor depending on prior background.

The outline for the full year is the following. Dr. Vasu Tewari will teach the third quarter.

- Fall = Enumeration: Every discrete process leads to questions of
existence, enumeration and optimization. This is the foundation of
Combinatorics. In this quarter we will present the basic
combinatorial objects and methods for counting various arrangements of
these objects.
- Basic counting methods.
- Sets, multisets, permutations, and graphs.
- Inclusion-exclusion.
- Recurrence relations and integer sequences.
- Generating functions.
- Partially ordered sets.
- Complexity Theory

- Winter = Graph Theory:
Graphs are among the most important
structures in combinatorics. They are universally applicable for
modeling discrete processes. We will introduce the fundamental
concepts and some of the major theorems. The existence questions of
combinatorics are prevalent in graph theory.
- Basic graph structures
- Matroids
- Matchings
- Planar graphs
- Colorings
- Ramsey Theory
- Random graphs and probabilistic methods
- Graph minor theorem

- Spring = Algebraic and Geometric Combinatorics:
This quarter will focus on
the topic of hyperplane arrangements. A hyperplane arrangement is a finite collection of codimension one subspaces in a finite-dimensional vector space over some field. The theory of hyperplane arrangements is a thriving field of research and its richness is demonstrated by the fact that it shows up in areas as diverse as topology, combinatorics and geometry. Surprisingly, many questions of geometric or topological flavor regarding hyperplane arrangement are essentially combinatorial questions. For the most part, this course aims at making the preceding statement precise before discussing some recent developments.
- Hyperplane arrangements and the associated intersection lattice
- The characteristic polynomial of a hyperplane arrangement
- The finite field method following Crapo-Rota and Athanasiadis
- Graphic arrangements and connections with chromatic polynomials
- Coxeter groups and reflection arrangements
- The topology of the complement of arrangements, and the Orlik-Solomon algebra
- Connections to zonotopes and oriented matroids
- Connections to diagonal harmonics

Textbooks:

Exercises: The single most important thing a student can do to learn combinatorics is to work out problems. This is more true in this subject than almost any other area of mathematics. Exercises will be assigned each week but many more good problems are to be found, with solutions, in your textbooks. Do as many of them as you can.

Problem sets: Grading will be based mostly on weekly problem sets due on Wednesdays. The problems will come from the text or from additional reading. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. We will have weekly problem sessions to discuss the harder problems. The time will be determined in the first week of class.

Grading Problem Sets: Students will be asked to grade another students problem set. More instructions to come with the first assignment.

Optional final: The final exam will be held in the two hour time slot assigned for our class period to be determined. The exam will include both in class work on an open problem and a written component due 24 hours later. More details will be given in class.

Optional presentations: Several recently published research articles will be presented in class by students. You will have the option of doing a presentation sometime this year. If you do a presentation, your lowest homework grade will be dropped.

Computing: Use of computers to verify solutions, produce examples, and prove theorems is highly valuable in this subject. Please turn in documented code if your proof relies on it. If you don't already know a computer language, then try SAGE, Maple, Mathematica or GAP. All three are available for free on our math department servers.

* Lecture 1: Introduction to combinatorial reasoning, counting functions, combinatorial objects. Generating functions. Read Chapter 1. PS# handed out.

* Lecture 2: Generating functions. Bijective proofs. Catalan numbers.

* Lecture 3: Basic counting principles. Sets, multisets, * compositions, and combinations

* Lecture 4: 10 ways to describe permutations. PS#2 handed out.

* Lecture 5: 2 more permutation representations: RSK and cycle notation. Stirling numbers of the 1st kind.

Sara Billey Last modified: Wed Sep 30 9:09:58 PDT 2015