Math 456/556: Networks and Combinatorics
Winter 2014, MWF 2:00 - 2:50, 303 Deady Hall
Professor Nicholas Proudfoot
Contact Information:
Email: njp@uoregon.edu
Office: 322 Fenton Hall
Office Hours: MWF 3:00 - 3:30 (after class) or by appointment.
Text: Richard A. Brualdi, Introductory Combinatorics, 5th edition, available at the Duck Store.
We will follow the book fairly closely, and you are encouraged to read ahead.
Prerequisites: You should have some experience writing proofs, at the level of Math 231, 261,
or 307. You will also need to know a little bit of differential calculus for the section on generating
functions (Week 6). Specifically, you should know the definition of the Taylor series of a function,
and you should be able to differentiate power series and rational functions.
Exams: There will be one mid-term examination on Friday, February 7th. The final exam will
be held on Monday, March 17th, at 3:15 pm. You may use a calculator if you want, though no
problems will require a calculator. You may not use the calculator on your phone or tablet (or
anything else that can connect to the network). You may also bring one page of notes to both the
mid-term and final exam.
Homework: Homework assignments are due at the beginning of class. Late assignments will not
be accepted without previous permission. Collaboration on homework assignments is allowed and
encouraged, provided that you write up your solutions independently. Copying another student’s
solutions, even if you worked on the problems together, is considered plagiarism.
Grading: Your score in the class will be determined by your mid-term exam (20%), final exam
(40%), and homework assignments (40%). Your lowest homework assignment score will be dropped.
If your final exam score is better than your mid-term exam score (or if you miss the mid-term exam),
then your mid-term score will be dropped, and your final will count for 60%. There will be no makeup exams. The correspondence between numerical scores and letter grades will be determined at
the end of the term.
Here is a week-by-week schedule. Dates for covering particular topics are approximate, but homework due dates and exam dates are firm.
Week 1. Counting things (Chapter 2)
This chapter is devoted to counting the number of different ways to arrange letters, numbers,
doughnuts, or other objects, subject to various constraints. This will involve finding the correct
combinatorial formulas to model real world problems (about doughnuts).
Homework #1 due Friday, 1/10. Chapter 2: 2, 3, 6, 10, 13, 15.
Week 2. The pigeonhole principle (Chapter 3)
The pigeonhole principle is both surprisingly simple and surprisingly powerful; applications will
include problems in number theory and Ramsey theory. You will be expected to write some proofs
for this homework assignment.
Homework #2 due Friday, 1/17.
Week 3. Binomial coefficients (Chapter 5)
We will prove everything you’ve ever wanted to know about binomial coefficients using various techniques. These will include mathematical induction, as well as the ubiquitous combinatorial trick of
“counting the same thing in two different ways”. This topic will also serve as an introduction to
generating functions.
Homework #3 due Monday, 1/27. (Week 3 is shortened by MLK Day, which is why the homework
will be due on Monday.)
Week 4. More counting (Chapter 6)
We will introduce the inclusion-exclusion principle, which will allow us to solve more sophisticated
counting problems. We will definitely cover Sections 6.1-6.3, and we will go beyond that if time
permits.
Homework #4 due Monday, 2/3.
Week 5. Mid-term exam
We will have a practice exam on Monday, go over the practice exam on Wednesday, and have the
real exam on Friday, 2/7.
Week 6. Recurrence relations and generating functions (Chapter 7)
In this section you will learn to solve recurrence relations, such as the one for the Fibonacci sequence: f0 = 0, f1 = 1, and fn = fn−1 + fn−2 for all n ≥ 2. Our main technique will be the
beautiful theory of generating functions. By the end, you should be able to solve an arbitrary
linear homogeneous recurrence relation, as well as some non-linear ones.
Homework #5 due Wednesday, 2/19. (This is a long topic that will probably spill over into Week
7, hence the extra time.)
Week 7. Graph theory (Chapter 11)
This topic will be an introduction to the theory of graphs, focusing on questions regarding the
existence of paths with various nice properties. At this point you will again be expected to write
some proofs.
Homework #6 due Wednesday, 2/26.
Week 8. More graph theory (Chapter 12)
A highlight of this section is the problem of counting the number of ways to color a map so that
neighboring “countries” have different colors.
Homework #7 due Wednesday, 3/5.
Week 9. Networks (Chapter 13)
This section is devoted to using graph theory to develop a mathematical model for real world networks (such as electrical circuits).
Homework #8 due Wednesday, 3/12 (note that this is during Dead Week).
Week 10. Final review
We will have a practice exam on Wednesday and go over it on Friday (the last day of class).