# 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.