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Combinatorics
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.
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.
Sites in Combinatorics
East Side, West Side
Lecture notes by Herbert S. Wilf on generating combinatorial objects and Maple.
The Cutting Sticks Problem
An unsolved combinatorial problem by Frans Faase.
CADCOM
Committee for the Advancement of Discrete and Combinatorial Mathematics.
British Combinatorial Committee
Conference listings, bulletin and links maintained by Peter Cameron.
The Combinatorics Net
Maintained by Bill Chen.
ArXiv Front: CO Combinatorics
Combinatorics section of the Front for the Mathematics ArXiv.
SIAM Activity Group on Discrete Mathematics
Provides an opportunity to unify pure discrete mathematics and areas of applied research.
Enumeration of the Binary Trees
A note on Catalan numbers by Ivan Galkin.
The Combinatorial Object Server
Select a type together with specific parameter values and COS will return a list of all such combinatorial objects.
Topics in Mathematics - Combinatorics
In the Mathematics Archive at UTK.
Extremal Combinatorics
With Applications in Computer Science by Stasys Jukna.
Encyclopedia of Combinatorial Structures
Searchable database of structures with specifications and counts.
Combinatorial Catalogues
Combinatorial objects such as graphs, designs and geometries collected by Gordon Royle.
Algebraic Combinatorics via Finite Group Actions
A hypertext by A. Betten, H. Fripertinger and A. Kerber.
CATS: Combinatorial Algorithms Test Sets
Searchable index of problems, links and methodology maintained by the Association for Computing Machinery.
Amazing Mathematical Object Factory
Generators for several basic combinatorial objects by Frank Ruskey and Scott Lausch.
A Survey of Venn Diagrams
Various facts and figures about Venn diagrams by Frank Ruskey.
G-Systems
Theory of chord structures to create music through algorithms by Vladimir Ladma.
Combinatorics
Subfields in the AMS MSC 2000 classification 05.
Combinatorial Data
Brendan McKay's collection of miscellaneous tables in graph6 or sparse6 format.
The Hyperbook of Combinatorics
A project to develop a hypertext on major topics in combinatorics.
On-line Dictionary of Combinatorics
An expanding web text by Joe Fields.
Catalogue of Lattices
A catalogue of different types with applications to packing spheres by Neil Sloane and Gabriele Nebe.
Combinatorial Mathematics Society of Australasia
Combinatorics throughout Australia, New Zealand and neighbouring countries.
The Math Forum Math Library - Combinatorics
Comprehensive catalog of websites relating to Combinatorics.
Math Pages - Combinatorics
About 40 "informal notes" on combinatorics by Kevin Brown.
Generating Functions
Introduction to generating functions with interactive java applets by Alexander Bogomolny.
Operational Research Programming Trivia
Computer code for research algorithms in operational research and combinatorial optimization. |
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