Combinatorics studies problems involving finite sets of objects that are defined by certain specified properties. For example, the objects in question may themselves be sets, numbers, graphs or other geometrical configurations. Enumerative combinatorics is concerned with counting the number of objects of a certain kind. Extremal combinatorics is concerned with finding the optimal objects of a certain kind. Topological methods, algebraic methods and even probabilistic methods have been used to solve combinatorial problems. Computer algorithms have also been used to solve some seemingly intractable combinatorial problems. Conversely, combinatorial methods have been used successfully to solve problems in many areas of mathematics and computer science. Here is a sample problem that would use combinatorics: Strangers and Acquaintances (F.P. Ramsey 1930): What is the least number of people that you need to have in a room so that there is always a group of three mutual strangers or a group of three mutual acquaintances? The answer is six.
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- The Combinatorics Net Maintained by Bill Chen.
- ArXiv Front: CO Combinatorics Combinatorics section of the Front for the Mathematics ArXiv.
- Enumeration of the Binary Trees A note on Catalan numbers by Ivan Galkin.
- Topics in Mathematics - Combinatorics In the Mathematics Archive at University of Tennessee, Knoxville.
- Extremal Combinatorics With Applications in Computer Science by Stasys Jukna.
- Encyclopedia of Combinatorial Structures Searchable database of structures with specifications and counts.
- Algebraic Combinatorics via Finite Group Actions A hypertext by A. Betten, H. Fripertinger and A. Kerber.
- Combinatorial Data Brendan McKay's collection of miscellaneous tables in graph6 or sparse6 format.
- The Math Forum Math Library - Combinatorics Comprehensive catalog of websites relating to Combinatorics.
- Generating Functions Introduction to generating functions with interactive java applets by Alexander Bogomolny.
- Series Expansions Iwan Jensen counts polyominoes (aka lattice animals), paths, and various related quantities.
- The Cutting Sticks Problem An unsolved combinatorial problem by Frans Faase.
- Combinatorics MathWorld article with basic definitions and links.
- Polya's Enumeration Theorem An article on generating functions and Polya's theory of counting by Karl Dahlke.
- Squared Squares Tilings of squares and rectangles, the classical problem and variations.
- Enumeration of Polyhedra Enumeration of polyhedra with up to 9 faces by Steven Dutch.
- A=B Downloadable combinatorics text by Marko Petkovsek, Herbert Wilf and Doron Zeilberger. Published by A. K. Peters.
- G-Systems Theory of chord structures to create music through algorithms by Vladimir Ladma.