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This definition of combinatorial mathematics to be found in Haider and Heise's book is certainly not the worst one. It implies in particular that combinatorics is not restricted to finite structures. But finiteness is always present and plays an essential role. Properties of finite sets are used permanently but usually unreflectedly: Proper subsets of a finite set have a smaller cardinal than the universe they belong to, injective mappings of a finite set into itself are surjective, and conversely. Moreover, finite sets admit essentially only one linear order. Finite topological spaces are always compact as their power set is finite.
Foreword
Hints for the reader
Dedekind Triples: The Fundamental Gauge
Finite Sets: The Basic Objects of Combinatorics
Familiar Realizations of Dedekind Triples
Adding without a Carry: Ein Glasperlenspiel
Rudiments of Universal Algebra
Embedding Commutative Semigroups into Groups: Localization
The Revolving Door Algorithm
Partitions of Finite Sets
Gray Codes: The General Case
A Little Bit on Graphs
Operator Groups
The Symmetric Groups: Combinatorial Properties
The Symmetric Groups: Algebraic Properties
Lyndon Words
Galois Fields: Counting Irreducible Polynomials
Ordered Sets
The Axiom of Choice: Equivalent Principles and Consequences
The Marriage Theorem
Independence Structures
Free Constructions
Symmetric Polynomials
Lie Algebras
Ordered Groups
The Fundamental Theorem of Algebra: A Final Highlight
Bibliography
Index of Algorithms and Procedures
Index