Summary:This three quarter topics course on Combinatorics includes Enumeration, Graph Theory, and Probabilistic 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:
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 text. 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 at 12:45pm to discuss the harder problems.
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 zeno and theano. Note, theano is our best set of processors for computing.
* Lecture 1: Introduction to combinatorial reasoning, counting functions, combinatorial objects. Generating functions. Read Chapter 1.
* Lecture 2: Generating functions. Catalan numbers.
* Lecture 3: Basic counting principles. Sets, multisets, * compositions, and combinations
* Lecture 4: 10 ways to describe permutations
* Lecture 5: Permutation statistics, Stirling numbers of the 1st * kind. Descents, Eulerian numbers and polynomials, q-analogs of n!.
* Lecture 6: Partitions of a sets vs. of an integer.
* Lecture 7: Permutation statistics for multisets, Guassian polynomials, and Grassmannians.
* Lecture 8: Paper presentation.
* Lecture 9: Bell numbers, Stirling numbers of 2nd kind and the * Twelvefold Way.
* Lecture 10: Inclusion-Exclusion Principle. Rook Theory.
* Lecture 11: Rook placements on Ferrers boards.
* Lecture 12: Unimodal Sequences and log concavity. Involutions.
* Lecture 13: Determinants and lattice path enumeration.
Lecture 14: Paper presentation.
* Lecture 15: Partially Ordered Sets, 6 nice properties of posets. Begin reading Chapter 3.
* Lecture 16: Lattices.
* Lecture 17: Distributive lattices and the FTFDL.
* Lecture 18: Chains in distributive lattices and the Mobius function.
* Lecture 19: Mobius Inversion. Finish reading Chapter 3.
* Lecture 20: Ten Fantastic Facts on Bruhat Order. Problem set 6 due.
* Lecture 21: Paper presentation.
* Lecture 22: Rational generating functions. Begin reading Chapter 4 and re-read Chapter 1 Section 1.1.
Lecture 23: Calculus of Finite Differences, polynomial and quasipolynomial sequences.
* Lecture 24: Video "N is a number; The life of Paul Erdos.
* Lecture 25: Paper Presentation
* Lecture 26: Plane Partitions
* Lecture 27: Complexity Theory/Algorithms
* Lecture 28: Complexity Theory/Algorithms
* Lecture 29: Paper presentation
Friday December 7, 2007 * Lecture 30: Conclusion and discussion of winter quarter.