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Combinatorics
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Combinatorics (or combinatorial analysis) is the study of countable (finite or countably infinite) discrete objects.
- Finitary combinatorics is the study of finite discrete objects.
- Infinitary combinatorics is the study of countably infinite discrete objects.
- Enumerative combinatorics is concerned with enumerating (in the sense of counting) the discrete objects of a given kind and size, rather than producing an explicit list. Examples of this type of problem are counting permutations and counting combinations.
- Extremal combinatorics (cf. combinatorial optimization) studies how large or how small a collection of finite objects can be, if it has to satisfy certain restrictions.
Contents
Techniques from other fields employed in various combinatorial contexts
- Algebraic combinatorics
- Employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts.
- Analytic combinatorics
- Employs analytical techniques in various combinatorial contexts.
- Geometric combinatorics
- Employs geometric techniques in various combinatorial contexts.
- Probabilistic combinatorics
- Employs probabilistic techniques in various combinatorial contexts.
- Topological combinatorics
- Employs topological techniques in various combinatorial contexts.
Combinatorial techniques employed in other fields
- Combinatorial algebra
- Employs combinatorial techniques in various algebraic contexts.
- Combinatorial game theory
- Employs combinatorial techniques in various game-theoretic contexts.
- Combinatorial geometry
- Employs combinatorial techniques in various geometric contexts.
- Combinatorial number theory
- Employs combinatorial techniques in various number-theoretic contexts.
- Combinatorial optimization (applied mathematics and theoretical computer science) (cf. extremal combinatorics)
- Employs combinatorial techniques to find an optimal ("largest", "smallest", or "optimal") object from a finite set of objects. In many such problems, exhaustive search is not feasible.
See also
