What is Combinatorics? | |
Example: Perfect Covers of Chessboards | |
Example: Magic Squares | |
Example: The Four-Color Problem | |
Example: The Problem of the 36 Officers | |
Example: Shortest-Route Problem | |
Example: Mutually Overlapping Circles | |
Example: The Game of Nim | |
The Pigeonhole Principle | |
Pigeonhole Principle: Simple Form | |
Pigeonhole Principle: Strong Form | |
A Theorem of Ramsay | |
Permutations and Combinations | |
Four Basic Counting Principles | |
Permutations of Sets | |
Combinations of Sets | |
Permutations of Multisets | |
Combinations of Multisets | |
Finite Probability | |
Generating Permutations and Combinations | |
Generating Permutations | |
Inversions in Permutations | |
Generating Combinations | |
Generating r-Combinations | |
Partial Orders and Equivalence Relations | |
The Binomial Coefficients | |
Pascal's Formula | |
The Binomial Theorem | |
Unimodality of Binomial Coefficients | |
The Multinomial Theorem | |
Newton's Binomial Theorem | |
More on Partially Ordered Sets | |
The Inclusion-Exclusion Principle and Applications | |
The Inclusion-Exclusion Principle | |
Combinations with Repetition | |
Derangements | |
Permutations with Forbidden Positions | |
Another Forbidden Position Problem | |
Möbius Inversion | |
Recurrence Relations and Generating Functions | |
Some Number Sequences | |
Generating Functions | |
Exponential Generating Functions | |
Solving Linear Homogeneous Recurrence Relations | |
Nonhomogeneous Recurrence Relations | |
A Geometry Example | |
Special Counting Sequences | |
Catalan Numbers | |
Difference Sequences and Stirling Numbers | |
Partition Numbers | |
A Geometric Problem | |
Lattice Paths and Schröder Numbers | |
Systems of Distinct Representatives | |
General Problem Formulation | |
Existence of SDRs | |
Stable Marriages | |
Combinatorial Designs | |
Modular Arithmetic | |
Block Designs | |
Steiner Triple Systems | |
Latin Squares | |
Introduction to Graph Theory | |
Basic Properties | |
Eulerian Trails | |
Hamilton Paths and Cycles | |
Bipartite Multigraphs | |
Trees | |
The Shannon Switching Game | |
More on Trees | |
More on Graph Theory | |
Chromatic Number | |
Plane and Planar Graphs | |
A 5-color Theorem | |
Independence Number and Clique Number | |
Matching Number | |
Connectivity | |
Digraphs and Networks | |
Digraphs | |
Networks | |
Matching in Bipartite Graphs Revisited | |
Pólya Counting | |
Permutation and Symmetry Groups | |
Burnside's Theorem | |
Pólya's Counting formula | |
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