Foundations of Combinatorics

Course Materials

• Problem Sets Due Wednesdays.
• Solutions
• Other Handouts and Reading Assignments

Interesting Web Sites

• Combinatorics Seminar at UW
• Electronic Journal of Combinatorics
• Generatingfunctionology by Herb Wilf
• 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.
• Sloane's On-Line Encyclopedia of Integer Sequences
• Wikipedia Dictionary of almost all terminology. This is a great teaching and learning tool.
• JSTOR Get papers from most AMS journals and other sources.
• Graphviz Package Graph drawing software. Here is my dot file for Bruhat order. (Note, I haven't tried it with the latest version of Graphviz yet.)

Syllabus

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:

• 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
• 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 very common in graph theory.
  • Basic graph structures
  • Matchings
  • Planar graphs
  • Colorings
  • Ramsey Theory
  • Graph minor theorem
• Symmetric Functions: The symmetric group, S_n, acts on polynomials in n variables simply by permuting the variables. Polynomials which are fixed under this action are called symmetric polynomials. If we consider the limit as n approaches infinity we get symmetric functions. The symmetric functions appear in many aspects of mathematics including algebra, topology, combinatorics, and geometry. The Schur functions are an important basis for symmetric functions. One key application of symmetric function theory using Schur functions is to the representation theory of S_n. Another key application of symmetric function theory is to the study of the cohomology ring of the Grassmannian Manifold. Recently, a generalization of Schur functions known as Macdonald polynomials have become a very hot topic in research; we will briefly cover their relationship with the n!-Theorem and Hilbert Schemes
  • Partitions and Tableaux
  • Elementary and Homogeneous Symmetric Functions
  • Schur Functions and the Littlewood-Richardson Rule
  • Character Theory of S_n
  • Representation theory of GL_n
  • Grassmannian Manifold
  • Macdonald Symmetric Functions

• Fall: Enumerative Combinatorics; Volume 1 by Richard Stanley, Cambridge Studies in Advanced Mathematics, 49, 1997.
• Winter: Graph Theory by Reinhard Diestel, Springer-Verlag, GTM vol. 173 Second Edition, 2000.
• Spring: Young tableaux: with applications to representation theory and geometry by William Fulton. London Mathematical Society Student Texts, vol 35. Cambridge University Press, 1997.

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.

Tentative Schedule:

* 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, Gaussian polynomials, and Grassmannians.

* Lecture 8: Paper presentation.

* Lecture 9: Bell numbers, Stirling numbers of 2nd kind and the * Twelve-fold 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: Hyperplane arrangements

* Lecture 28: Complexity Theory/Algorithms

* Lecture 29: Paper presentation

* Lecture 30: Conclusion and discussion of winter quarter.