Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

(TO DO: organize, cross-reference, illustrate.)

Description

Aspects of combinatorics include:

• Counting the structures of a given kind and size (enumerative combinatorics)

• Deciding when certain criteria can be met

• Constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory)

• Finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization)

• Studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics)

Combinatorial problems arise in many areas of pure mathematics, notably:

• Algebra
• Probability theory
• Topology
• Geometry

Combinatorics has many applications in:

• Mathematical optimization
• Computer science
  • Obtain formulas and estimates in the analysis of algorithms
• Ergodic theory
• Statistical physics

Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context.

In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.

Graph theory

One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas.

See also

• Geometry
• Graph theory
• Mathematics

External links

• Combinatorics @ Wikipedia