Foundations of Combinatorics
Fall, 2021
Prof. Sara Billey
Monday, Wednesday, Friday 12:301:20
Padelford
C36 (inperson)
Class Size Limit
This class was over enrolled so now students will need an add code to
join the class. The priority will go to currently enrolled graduate
students in the math department. Please email both the instructor and
the advising office if are you in the priority group and hope to enroll.
Due to the high demand and covid
precautions, no one will be allowed to audit the class this quarter.
Course Materials
Interesting Web Sites
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.
 Inclusionexclusion.
 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 halfspaces in Rd), yet they have a very rich and continuously developing theory. While 3dimensional 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 ddimensional centrally symmetric polytope have at least 3d faces? Can the number of idimensional faces of a polytope be smaller than both the number of its vertices and the number of its topdimensional 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 Hpolytopes and Vpolytopes, the face lattice of a polytope.
 Graphs of polytopes  "classical" results: Steinitz's theorem for 3polytopes; 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 DehnSommerville relations; the upper and the lower bound theorems; the gtheorem and its consequences; face numbers of centrally symmetric polytopes.
 Gale diagrams and various counterexamples: polytopes with few
vertices, nonrational polytopes, nonpolytopal 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 JacobiTrudi identity, quasisymmetric functions;
 Young tableaux, the hook length formula, the RSK correspondence, growth diagrams, LittlewoodRichardson rules, jeu de taquin;
 representation theory of the symmetric group, the Frobenius characteristic map, the MurnaghanNakayama rule, SchurWeyl 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:
(there is some new suggested
language that I support  SB)
 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/religiousaccommodationspolicy/). Accommodations
must be requested within the first two weeks of this course using
the Religious Accommodations Request form
(https://registrar.washington.edu/students/religiousaccommodationsrequest/).”

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,
attentionrelated, learning, vision, hearing, physical or health
impacts), you are welcome to contact DRS at 2065438924 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."
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Sara Billey
Last modified: Mon Sep 27 12:59:00 PDT 2021