Math 274 - Combinatorics 

Math 274 is an introduction to combinatorics, the art of counting. Besides
being an interesting area of pure mathematics, combinatorics also has
applications to the analysis of algorithms in computer science, and to basic
probability theory. The tools we use, particularly generating functions,
occur in other parts of mathematics such as statistics. There are surprising
connections to mathematical analysis (formal power series) and to algebra
(group actions when counting objects with symmetries), although no particular
background in these areas will be assumed. 

In this course we cover five fundamental counting techniques. First, we start
with the basic theory of permutations and combinations, subject to various
restrictions. Second, we examine how power series can be used to represent
counting information in the form of generating functions, and how
manipulation of these generating functions can solve counting problems.
Third, we look at recurrence relations, where the number of objects of a
given size can be expressed in terms of the numbers of objects of smaller
sizes. Fourth, we see how the theory of inclusion-exclusion can be used to
count objects with combinations of properties. Finally, we apply group theory
to count objects with symmetry in Polya's Theorem. 

The course is suitable for graduate students and upper-level undergraduates.