Two sessions / week; 1 hour / session
The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research.
Every week, the first session is for lectures on different techniques and results related to bijective proofs. Problems of varying levels of difficulty (some unsolved as the course progresses) related to the lecture will be passed out on the first session. In every second session, students will report on their work on the problems. All students must participate in this process. "Reasonable" collaboration on problem sets is permitted, but you shouldn't simply copy someone else's solution or a solution from an outside source.
There will be no text required for the course. All material will be based on the handouts. There is currently one book in print devoted solely to bijective proofs:
Stanton, D., and D. E. White. Constructive Combinatorics. This book is rather sophisticated.
A more elementary book by A. T. Benjamin, and J. J. Quinn, Proofs That Really Count, will soon be published, but it is not yet available.