# Foundations of Combinatorics

### Spring, 2012 Prof. Sara Billey

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

### 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
• Algebraic Combinatorics: 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 Schur functions is to the study of the cohomology ring of the Grassmannian Manifold. A third key application of Schur functions is to the representation theory of \$GL_{n}\$. We will survey the algebraic side of combinatorics while exploring these connections. Topics include:
• 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, Flag manifolds and their subvarieties
• Schubert Calculus
Textbooks:

Other Useful References and Textbooks:

• Representation Theory: A First Course by Bill Fulton and Joe Harris, Springer 1991. Also updated in 5th printing.
• Symmetric Functions and Hall Polynomials by Ian Macdonald, Oxford Mathematical Monographs 1975,1995.
• The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions by Bruce Sagan, Springer, 1991, 2001.
• Enumerative Combinatorics; Volume 2 by Richard Stanley, Cambridge Studies in Advanced Mathematics, 62, 1999. Available in paperback.
Student presentations: A graduate level Combinatorics course is surprisingly close to the frontier of research in this area. Each student will present a recent journal article to be chosen with the instructor. Presentations will occur during the last 2 weeks of the quarter. The presentation will count toward 50% of the grade.

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, within, in your textbooks. Do as many of them as you can.

Problem sets: The other 50% of the grade will be based on weekly problem sets due on Wednesdays. The problems be assigned during the course of each lecture. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. We will have a problem session on Monday or Tuesday afternoons to discuss the harder problems. The time will be determined in the first week of class.

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.

### Schedule:

(T) indicates tentative.

* Lecture 1: Historical approach to Schur functions. The first of 4 definitions to be given.

* Lecture 2: Jacobi-Trudi Identity

* Lecture 3: Pieri's formula and the column strict tableaux definition of Schur functions

* Lecture 4: Robinson-Schensted-Knuth correspondence, Cauchy identities

* Lecture 5: Jeu da taquin

* Lecture 6: Knuth Equivalence

* Lecture 7: Littlewood Richardson coefficients

* Lecture 8: Many variations on Littlewood-Richardson rules

* Lecture 9: Frame-Robinson-Thrall hook length formula, Edelman-Greene correspondence

* Lecture 10: (T) Specht modules

* Lecture 11: (T) Straightening algorithm

* Lecture 12: (T) Characters of the irreducible S_n representations and the Murnaghan-Nakayama rule

* Lecture 13: (T) GL_n representations

Lecture 14: (T) Characters and Schur functions

* Lecture 15: (T) Ideal of quadratic relations

* Lecture 16: (T) Grassmannians and other Flag varieties

* Lecture 17: (T) Plücker relations and the ideal of quadratic relations revisited

* Lecture 18: (T) Schubert polynomials, rc-graphs

* Lecture 19: (T) Matrix Schubert varieties

* Lecture 20: (T) Richardson varieties (and Positroids if possible or maybe in a student presentation)

* Lecture 21: (T) Quantum Schubert polynomials

* Lecture 22: (T) Criteria for smoothness in Schubert varieties

Lecture 23: (T) Other pattern avoidance criteria for Schubert varieties

* Lecture 24: Student presentation

* Lecture 25: Student presentation

* Lecture 26: Student presentation

* Lecture 27: Student presentation

* Lecture 28: Student presentation

* Lecture 29: Student presentation

* Lecture 30: (T) Survey of open problems

Sara Billey
Last modified: Tue Apr 10 15:36:25 PDT 2012