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Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics), with deciding when the criteria can be met, with constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and with 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 part of combinatorics is graph theory which is now connected to other areas.
An example of a combinatorial question is the following: What is the number of possible orderings of a deck of 52 playing cards? That number equals 52! (fifty-two factorial). It may seem surprising that this number (80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000) is so big. Comparing that number to some other large numbers, it is greater than the square of Avogadro's number, 6.022 × 1023.
More on [ Combinatorics ]
Algorithms :: Computers
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A Survey of Venn Diagrams - Various facts and figures about Venn diagrams by Frank Ruskey.
Meta Description: [ Venn Diagram Survey: combinatorial facts, beautiful figures, n=3 and beyond ]
Algebraic Combinatorics via Finite Group Actions - A hypertext by A. Betten, H. Fripertinger and A. Kerber.
ArXiv Front: CO Combinatorics - Combinatorics section of the Front for the Mathematics ArXiv.
Catalogue of Lattices - A catalogue of different types with applications to packing spheres by Neil Sloane and Gabriele Nebe.
CATS: Combinatorial Algorithms Test Sets - Searchable index of problems, links and methodology maintained by the Association for Computing Machinery.
Combinatorial Catalogues - Combinatorial objects such as graphs, designs and geometries collected by Gordon Royle.
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Combinatorial Data - Brendan McKay's collection of miscellaneous tables in graph6 or sparse6 format.
Combinatorics - Subfields in the AMS MSC 2000 classification 05.
Combinatorics - MathWorld article with basic definitions and links.
East Side, West Side - Lecture notes by Herbert S. Wilf on generating combinatorial objects and Maple.
Encyclopedia of Combinatorial Structures - Searchable database of structures with specifications and counts.
Meta Description: [ (none) ]
Enumeration of the Binary Trees - A note on Catalan numbers by Ivan Galkin.
Extremal Combinatorics - With Applications in Computer Science by Stasys Jukna.
Meta Description: [ Web page supporting the book Extremal Combinatorics: list of misprints, further exercises and problems, links, etc. ]
G-Systems - Theory of chord structures to create music through algorithms by Vladimir Ladma.
Meta Description: [ An algebraic theory of chord structures ]
Generating Functions - Introduction to generating functions with interactive java applets by Alexander Bogomolny.
Lectures in Combinatorics - An introduction to partially ordered sets, Dilworth's theorem, lattices and combinatorial geometries.
Meta Description: [ An introduction to partially ordered sets, Dilworth's theorem, lattices and combinatorial geometries. ]
On-line Dictionary of Combinatorics - An expanding web text by Joe Fields.
Operational Research Programming Trivia - Computer code for research algorithms in operational research and combinatorial optimization.
Meta Description: [ %if(,,www.ortrivia.com is coming soon!) ]
Series Expansions - Iwan Jensen counts polyominoes (aka lattice animals), paths, and various related quantities.
The Combinatorial Object Server - Select a type together with specific parameter values and COS will return a list of all such combinatorial objects.
Meta Description: [ The (Combinatorial) Object Server: A mathematical tool for producing lists of combinatorial objects. ]
The Combinatorics Net - Maintained by Bill Chen.
The Cutting Sticks Problem - An unsolved combinatorial problem by Frans Faase.
The Math Forum Math Library - Combinatorics - Comprehensive catalog of websites relating to Combinatorics.
Meta Description: [ The Math Forum's Internet Math Library is a
comprehensive catalog of Web sites and Web pages relating to the study
of mathematics. This page contains sites relating to Combinatorics. ]
Topics in Mathematics - Combinatorics - In the Mathematics Archive at UTK.
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