Suppose we have a set of N objects that have various properties α, β, γ, … λ. Each of the objects may any or none of the properties. Let Nα be the number of objects that have property α. Some of these objects may have other properties in addition to property α; that doesn’t matter. (In fact, that’s the whole idea!) Similarly, let Nβ be the number of objects that have property β, and so on.
January 5. In his first lecture, Pólya discussed in general terms what combinatorics is about: The study of counting various combinations or configurations. He stated with a problem based on the mystical sign known, appropriately, as an “abracadabra”.
February 28. Given a set of men and a set of women, a matching is a set of pairs, each pair containing one man and one woman, such that no person is in more than one pair. We shall be interested in finding matchings satisfying various criteria. The first problem we’ll consider is called the stable marriage problem. We assume that there are the same number of men as women, and that each person ranks the people of the opposite sex in order of preference.
February 23. This was Tarjan’s first lecture, and he started by announcing what he intended to cover during his seven lectures. Specifically, he said he would be discussing miscellaneous problems in existential and constructive combinatorics, with the emphasis on the constructive side. This chapter, on the other hand, deals almost entirely with existential combinatorics, although some of the proofs are constructive.
February 2. Pólya’s title for this part of the course was actually “Counting Configuration Non-Equivalent with Respect to a Given Permutation Group”. Just about everybody else refers to it as “Pólya’s Theory of Counting”. This later title is somewhat easier to remember, though not as indicative of the content. We’ll stick to the simpler title; for the content, read on!
Network flow problems are usually presented in terms of directed graphs. A directed graph is the same as the graphs we've been working with, except that each edge is assigned a direction. Another way of stating the distinction is that, whereas in an undirected graph each edge is a pair of vertices, in a directed graph each edge is an ordered pair of vertices.
This chapter contains the midterm exam and the solutions thereto. The exam was open book and “take home”; students were given one week to work on it. They were advised they would find the exam somewhat “Open-ended”. They were not required to do problems 1c and 2b, though they were strongly encouraged to do so. It was stated that extra credit would be given to students who attempted those problems.
