Foundations of Combinatorics

Class Size Limit

Course Materials

• Enumerative Combinatorics: Volume 1 by Richard Stanley (2012). A pdf of a draft of the textbook and the errata can be found here.
• Algebraic and geometric methods in enumerative combinatorics by Federico Ardila, in the "Handbook of Enumerative Combinatorics" edited by Miklos Bona. (2015).
• Combinatorics: The Art of Counting by Bruce Sagan (2020).
• Problem Sets due on Wednesdays.
• Selected Solutions
• Other Handouts and Reading Assignments
• Some Reading Assignments from Past Years

Interesting Web Sites

• Combinatorics Seminar at UW
• Recent preprints on research in Combinatorics from the arXiv.
• Mathscinet Index to all published research in mathematics.
• On-Line Encyclopedia of Integer Sequences (OEIS) Amazing collection of integer sequences started by Neil Sloane. Try to "listen" to a sequence! Plot two sequences for comparison. Or "browse" the Best Sequences. As of September 27 15:24 EDT 2021 the OEIS contains 347691 sequences.
• FindStat is also a great tool for studying statistics on combinatorial objects.
• Generatingfunctionology by Herb Wilf
• Wikipedia Dictionary of almost all terminology. This is a great teaching and learning tool. Check out the extensive article on Combinatorics.
• SAGE: Open Source Mathematical Software A collection of mathematical tools to do symbolic computation, combinatorial algorithms, graph manipulation, exact linear algebra, etc. Plus, was partially developed right here at UW!
• Graphviz Package Graph drawing software. Here is my dot file for Bruhat order.

Syllabus

Summary: This three quarter topics course on Combinatorics includes Enumeration, Polytopes, and Algebraic 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 this quarter is the following:

• Fall = Enumeration (Billey): 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

• Winter = Polytopes (Prof. Isabella Novik): Polytopes are very simple objects (defined as the convex hull of a finite set of points in Rd, or equivalently as a bounded intersection of finitely many closed half-spaces in Rd), yet they have a very rich and continuously developing theory. While 3-dimensional polytopes are well understood, a lot of facts that are obvious in dimension three are either false or unknown in higher dimensions. (For instance, does every d-dimensional centrally symmetric polytope have at least 3d faces? Can the number of i-dimensional faces of a polytope be smaller than both the number of its vertices and the number of its top-dimensional faces? These questions are still open except for a few small values of d.) In this course, we will mainly concentrate on certain combinatorial and geometric aspects of polytopes. Specifically, depending on time and interest, we'll discuss some of the following topics:
  • Introduction to polytopes: convex sets in general (separation theorem, Radon's and Caratheodory's theorems); polarity, equivalence of H-polytopes and V-polytopes, the face lattice of a polytope.
  • Graphs of polytopes -- "classical" results: Steinitz's theorem for 3-polytopes; Balinski's theorem; reconstructing simple polytopes from their graphs.
  • Graphs of polytopes -- this century results: Santos's counterexample to the Hirsch conjecture; positive results towards the polynomial Hirsch conjecture.
  • Face numbers of simplicial polytopes: the Dehn-Sommerville relations; the upper and the lower bound theorems; the g-theorem and its consequences; face numbers of centrally symmetric polytopes.
  • Gale diagrams and various counterexamples: polytopes with few vertices, non-rational polytopes, non-polytopal spheres.
• Spring = Algebraic Combinatorics (Prof. Ricky Liu): Algebraic combinatorics is the study of the interaction between algebraic objects, such as rings and group representations, and combinatorial objects, such as permutations and tableaux. This course will cover three closely related areas-- the ring of symmetric functions, the combinatorics of Young tableaux, and the representation theory of the symmetric group-- and highlight the connections between them. Topics will include:
  • the five bases of symmetric functions, Hall inner product, the Cauchy identity, the involution w, the Jacobi-Trudi identity, quasisymmetric functions;
  • Young tableaux, the hook length formula, the RSK correspondence, growth diagrams, Littlewood-Richardson rules, jeu de taquin;
  • representation theory of the symmetric group, the Frobenius characteristic map, the Murnaghan-Nakayama rule, Schur-Weyl duality.
  • Additional topics may be covered depending on time and interest.

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. Each problem is worth 10 points. Different problem sets may have a differnt number of problems. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. Write up all of your solutions on your own! 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 sometimes be asked to grade another students problem set. More instructions to come with the first assignment.

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 Python, SAGE, Maple, Mathematica or GAP. All are available for free on our math department servers. See the UW Math Computing wiki for more details.

Tentative Schedule:

* Lecture 1: Introduction to combinatorial reasoning, counting functions, combinatorial objects. Generating functions. Read Chapter 1 of EC 1 and other sources. PS#1 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.

University Policy and Resources:

• Face Coverings in the Classroom: The health and safety of the University of Washington community are our top priorities. Please review and adhere to the UW COVID Face Covering Policy.

• Religious Accommodations Policy: “Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW's policy, including more information about how to request an accommodation, is available at Religious Accommodations Policy (https://registrar.washington.edu/staffandfaculty/religious-accommodations-policy/). Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form (https://registrar.washington.edu/students/religious-accommodations-request/).”

• Access and Accommodations: "Your experience in this class is important to me. If you have already established accommodations with Disability Resources for Students (DRS), please communicate your approved accommodations to me at your earliest convenience so we can discuss your needs in this course. If you have not yet established services through DRS, but have a temporary health condition or permanent disability that requires accommodations (conditions include but not limited to; mental health, attention-related, learning, vision, hearing, physical or health impacts), you are welcome to contact DRS at 206-543-8924 or uwdrs@uw.edu or disability.uw.edu. DRS offers resources and coordinates reasonable accommodations for students with disabilities and/or temporary health conditions. Reasonable accommodations are established through an interactive process between you, your instructor(s) and DRS. It is the policy and practice of the University of Washington to create inclusive and accessible learning environments consistent with federal and state law."

• Student Conduct and Academic Integrity Code: "The University of Washington Student Conduct Code (WAC 478-121) defines prohibited academic and behavioral conduct and describes how the University holds students accountable as they pursue their academic goals. Allegations of misconduct by students may be referred to the appropriate campus office for investigation and resolution. More information can be found online at https://www.washington.edu/studentconduct/"

• Safety: "Call SafeCampus at 206-685-7233 anytime – no matter where you work or study – to anonymously discuss safety and well-being concerns for yourself or others. SafeCampus's team of caring professionals will provide individualized support, while discussing short- and long-term solutions and connecting you with additional resources when requested."