Summary:This three quarter topics course on Combinatorics includes Enumeration, Graph Theory, and Algebraic Combinatorics. Combinatorics has connections to all areas of mathematics 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 basic knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research. The outline for each quarter is the following:
In this class we will look at these two closely related fields. We will touch on many branches of mathematics including graph theory, probability, topology and geometric group theory. While all of these are useful, none of them are prerequisites.
There is no text but resources include Vic Reiner's notes on "Hopf Algebras In Combinatorics", Federico Ardila's lectures on "Hopf Algebras" and the book ``Coxeter Groups and Hopf Algebras'' by Marcelo Aguiar and Swapneel Mahajan.
Course requirements: a strong background in algebra on par with our Math 504/5/6 and basic enumerative combinatorics on par with 461/462. No prior knowledge of Hopf algebras will be assumed.
Additional Reading: A graduate level Combinatorics course is surprisingly close to the frontier of research in this area. Approximately, five recent journal articles will be handed out which are related to the current material. Occasionally, problems on the homework will related to this reading.
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 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 a problem session on Monday afternoons 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 every other week. More instructions to come with the first assignment.
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 or GAP. All three are installed on abel and theano.
* 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. Additional Reading: A Simple Proof of the Hook Length Formula by Kenneth Glass and Chi-Keung Ng. Source: The American Mathematical Monthly, Vol. 111, No. 8 (Oct., 2004), pp. 700-704