MacGillivray G. Combinatorial Mathematics

Sydney: University of Victoria, 2011. — 124 p.Factorials and Binomial Coefficients
Counting Principles
Introduction to Combinatorial Arguments
Exercises
The Binomial Theorem and Friends
The Binomial Theorem
Some Standard Combinatorial Arguments
Bertrand’s Ballot Problem
The Multinomial Theorem
Exercises
Advanced Counting Numbers
Stirling Numbers of the First Kind
Stirling Numbers of the Second Kind
Bell Numbers
Partitions
The Twelvefold Way
Derangements
Exercises
Generating Functions
Ordinary Power Series Generating Functions
Catalan Numbers
Triangulations of convex n-gons
Binary trees
Arranging symbols
– Correspondences
Partitions
Stirling Numbers of the Second Kind
Exponential Generating Functions
Onto Functions
Bell Numbers
Derangements
Exercises
PIE and Generalizations
The Principle of Inclusion and Exclusion (PIE)
Three Classical PIE Examples
Onto Functions and Stirling Numbers of the Second Kind
Derangements
Euler’s φ Function
In Exactly m Sets or In At Least m Sets
Rook Polynomials (Arrangements with Forbidden Positions)
Exercises
Ramsey Theory
The Pigeonhole Principle
Two-Colour Graph Ramsey Numbers
Multicolour Graph Ramsey Numbers
Hypergraphs and Ramsey’s Theorem
Ramsey-like Theorems
Convex m-gons
Arithmetic Progressions
Exercises
Polya Theory
Orbits and Stabilizers
Burnside’s Lemma
Conjugacy Classes
Equivalence Classes of Functions
Polya’s Theorem: The General Case
Exercises