Combinatorics

Preliminaries

 * What is Combinatorics?
 * Motivating Examples and Problems
 * Counting
 * Subsets of a set-The Binomial Coefficient
 * Binomial Theorem
 * Congruences

The Pigeonhole Principle

 * The Pigeon Hole Principle

Pairing problem

 * General principles
 * P. Hall's selection theorem
 * Applications to Latin squares and to coverings by dominoes of pruned chessboards.

The inclusion-exclusion principal

 * Applications to derangements
 * Applications to counting problems
 * Applications to rook polynomials

Partitions

 * Counting various types of partitions
 * Ferrers graphs
 * Self-conjugate partitions

Symmetric functions (and anti-symmetric functions)

 * Monomial symmetric functions
 * Elementary symmetric functions
 * Theory of equations
 * Newton's formulae and relations between symmetric functions
 * Indexing of symmetric functions by partitions.

Ramsey Theory

 * Ramsey's Theorem
 * Bounds for Ramsey numbers
 * Schur's Theorem