Foundations of Combinatorics

Fall, 2021

Prof. Sara Billey

Monday, Wednesday, Friday 12:30-1:20

Padelford C-36 (in-person)

Class Size Limit

This class was over enrolled so now students will need an add code to join the class. The priority will go to currently enrolled graduate students in the math department. Please email both the instructor and the advising office if are you in the priority group and hope to enroll. Due to the high demand and covid precautions, no one will be allowed to audit the class this quarter.

Course Materials

Interesting Web Sites


Summary: This three quarter topics course on Combinatorics includes Enumeration, Polytopes, and Algebraic 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 this quarter is the following:

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. Each problem is worth 10 points. Different problem sets may have a differnt number of problems. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. Write up all of your solutions on your own! 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 sometimes be asked to grade another students problem set. More instructions to come with the first assignment.

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 Python, SAGE, Maple, Mathematica or GAP. All are available for free on our math department servers. See the UW Math Computing wiki for more details.

Tentative Schedule:

* Lecture 1: Introduction to combinatorial reasoning, counting functions, combinatorial objects. Generating functions. Read Chapter 1 of EC 1 and other sources. PS#1 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.

University Policy and Resources:

(there is some new suggested language that I support -- SB)
Sara Billey
Last modified: Mon Sep 27 12:59:00 PDT 2021