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

Aspects of combinatorics include:

  • enumerative combinatorics: counting the structures of a given kind and size
  • combinatorial designs and matroid theory: deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria
  • extremal combinatorics and combinatorial optimization: finding "largest", "smallest", or "optimal" objects
  • algebraic combinatorics:
    • studying combinatorial structures arising in an algebraic context
    • applying algebraic techniques to combinatorial problems