Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Aspects of combinatorics include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and 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), and finding algebraic structures these objects may have (algebraic combinatorics).

Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain estimates on the number of elements of certain sets.

A mathematician who studies combinatorics is often referred to as a combinatorialist or combinatorist.

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• Computers: Algorithms (318)
• Science: Math: Algebra: Group Theory (33)
• Science: Math: Probability (98)

• A New Proof of the Four Colour Theorem
  dharwadker.org
  By Ashay Dharwadker. Includes profile, research papers for other algorithms, lecture notes, and student database.
• Combinatorial Data
  cs.anu.edu.au
  Brendan McKay's collection of miscellaneous tables in graph6 or sparse6 format.
• Generating Functions
  cut-the-knot.org
  Introduction to generating functions with interactive java applets by Alexander Bogomolny.