Combinatorics (Math 513) Textbook contents

METHODS

• Basic methods
• When we add and when we subtract
• When we multiply
• When we divide
• Applications of basic counting principles
• The pigeonhole principle
• Applications of basic methods
• Multisets and compositions
• Set partitions
• Partitions of integers
• The inclusion-exclusion principle
• The twelvefold way
• Generating functions
• Power series
• Warming up: Solving recurrence relations
• Products of generating functions
• Compositions of generating functions
• A different type of generating functions

TOPICS

• Counting permutations
• Eulerian numbers
• The cycle structure of permutations
• Cycle structure and exponential generating functions
• Inversions
• Advanced applications of generating functions to permutation enumeration
• Counting graphs
• Trees and forests
• Graphs and functions
• When the vertices are not freely labeled
• Graphs on colored vertices
• Graphs and generating functions
• Extremal combinatorics
• Extremal graph theory
• Hypergraphs
• Something is more than nothing: Existence proofs

AN ADVANCED METHOD

• Analytic combinatorics
• Exponential growth rates
• Polynomial precision
• More precise asymptotics

SPECIAL TOPICS

• Symmetric structures
• Designs
• Finite projective planes
• Error-correcting codes
• Counting symmetric structures
• Sequences in combinatorics
• Unimodality
• Log-concavity
• The real zeros property
• Counting magic squares and magic cubes
• A distribution problem
• Magic squares of fixed size
• Magic squares of fixed line sum
• Why magic cubes are different

APPENDIX

• The method of mathematical induction
• Weak induction
• Strong induction