Combinatorics
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Preliminaries
- What is Combinatorics?
- Motivating Examples and Problems
- Counting
- Subsets of a set-The Binomial Coefficient
- Binomial Theorem
- Congruences
The Pigeonhole 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 polynomialss
Linear recurrence relations
Generating functions
Catalan numbers
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.
