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