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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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Links:

East Side, West Side Lecture notes by Herbert S. Wilf on generating combinatorial objects and Maple. www.cis.upenn.edu/~wilf/lecnotes.html

The Combinatorics Net Maintained by Bill Chen. www.combinatorics.net

ArXiv Front: CO Combinatorics Combinatorics section of the Front for the Mathematics ArXiv. front.math.ucdavis.edu/math.CO

Enumeration of the Binary Trees A note on Catalan numbers by Ivan Galkin. ulcar.uml.edu/~iag/CS/Catalan.html

The Combinatorial Object Server Select a type together with specific parameter values and COS will return a list of all such combinatorial objects. www.theory.csc.uvic.ca/~cos

Topics in Mathematics - Combinatorics In the Mathematics Archive at UTK. archives.math.utk.edu/topics/combinatorics.html

Extremal Combinatorics With Applications in Computer Science by Stasys Jukna. www.thi.informatik.uni-frankfurt.de/~jukna/EC_Book

Encyclopedia of Combinatorial Structures Searchable database of structures with specifications and counts. algo.inria.fr/encyclopedia

Algebraic Combinatorics via Finite Group Actions A hypertext by A. Betten, H. Fripertinger and A. Kerber. www.mathe2.uni-bayreuth.de/frib/html/book/hyl00.html

CATS: Combinatorial Algorithms Test Sets Searchable index of problems, links and methodology maintained by the Association for Computing Machinery. www.jea.acm.org/CATS

A Survey of Venn Diagrams Various facts and figures about Venn diagrams by Frank Ruskey. www.combinatorics.org/Surveys/ds5/VennEJC.html

G-Systems Theory of chord structures to create music through algorithms by Vladimir Ladma. www.sweb.cz/vladimir_ladma/english/music/articles/dide99.htm

Combinatorics Subfields in the AMS MSC 2000 classification 05. www.ams.org/msc/05-xx.html

Combinatorial Data Brendan McKay's collection of miscellaneous tables in graph6 or sparse6 format. cs.anu.edu.au/~bdm/data

The Hyperbook of Combinatorics A project to develop a hypertext on major topics in combinatorics. www.combinatorics.net/hyper

Catalogue of Lattices A catalogue of different types with applications to packing spheres by Neil Sloane and Gabriele Nebe. www.research.att.com/~njas/lattices

The Math Forum Math Library - Combinatorics Comprehensive catalog of websites relating to Combinatorics. mathforum.org/library/topics/combinatorics

Math Pages - Combinatorics About 40 "informal notes" on combinatorics by Kevin Brown. www.mathpages.com/home/icombina.htm

Generating Functions Introduction to generating functions with interactive java applets by Alexander Bogomolny. www.cut-the-knot.org/blue/GeneratingFunctions.shtml

Operational Research Programming Trivia Computer code for research algorithms in operational research and combinatorial optimization. www.ortrivia.com

The Cutting Sticks Problem An unsolved combinatorial problem by Frans Faase. home.planet.nl/~faase009/cutsticks.html

Combinatorial Catalogues Combinatorial objects such as graphs, designs and geometries collected by Gordon Royle. www.csse.uwa.edu.au/~gordon/data.html

On-line Dictionary of Combinatorics An expanding web text by Joe Fields. www.southernct.edu/~fields/comb_dic

Four Colour Theorem: A Brief Historical Insight An essay on the history of the four color theorem by Dominic Verderaime. csci.adams.edu/~verderaimedj/FinalEssay

Series Expansions Iwan Jensen counts polyominoes (aka lattice animals), paths, and various related quantities. www.ms.unimelb.edu.au/~iwan/Series.html