Introduction to Combinatorics
Combinatorics is an area of mathematics that deals with the study of combination, enumeration, and permutations of sets of elements. It can be loosely defined as the mathematics of counting, as a result, algebra and its tools are essential to combinatorics. According to Mathematical Atlas, particular areas of study include graph and group theory (study of association schemes in strongly regular graphs and symmetry groups), Young tableaux, and coding theory (particularly in nonlinear codes).
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The Facts
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Combinatorics is the area of mathematics that focuses on the study of finite families of sets with particular subset arrangements. This study includes such topics as find the number of all possible combinations and permutations of a finite set. According to Mathematical Atlas, combinatorics is often introduced in combination with other elementary topics (such as probability and simple number theory). More advanced topics cover more sophisticated methods of counting sets (such as power series that form generating functions).
Features
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Other areas of enumerable combinatorics include asymptotic functions such as the estimates for a number of partitions of an integer and synthetic counting using umbral calculus (the study of seemingly unrelated polynomial equations). Designs (symmetric and asymmetric arrangements of certain sets and subsets) is the study of non-enumerable branch of combinatorics, according to Mathematical Atlas. Some famous problems include the Fano plane (seven points each falling into seven lines) and Latin squares (rectangular array of arranged elements with no respect to rows and columns).
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Function
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Combinatorics has many functions and purposes in mathematics. According to Mathematical Atlas, combinatorial arguments are used to determine coefficients to deduce function identities (such as the Ramanujan identities and other infinite sums and products). Combinatorics is also used to study Matroids (generalized geometries).
History
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The history of combinatorics can be traced back to Leonhard Euler, a Swiss mathematician working in Russia. According to Science, Euler made the beginning contributions to combinatorics in 1736 in both graph theory and enumeration. He studied movement along bridges and wrote a paper called "The Seven Bridges of Königsberg." Königsberg was a city on an island that had seven bridges. People from the town questioned whether it would be possible to "take a walk through town, starting and ending at the same place, and cross each bridge exactly once?" Euler studied this problem mathematically using points and lines, thus introducing the world to elementary graph theory.
Applications
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Combinatorics has many answers to pressing mathematical questions. Besides the previously mentioned applications to number, group, and graph theory, combinatorics is also important to geometry and algebraic topology. According to Mathematical Atlas, combinatorics is essential to elementary probability, statistics (in particular to the subtopic of experimental design), computational geometry, game theory, operational research, and computer science (in particular to combinatorial optimization).
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