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.
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.