Difference between revisions of "Combinatorics"
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| − | '''Combinatorics''' is the study of counting. | + | '''Combinatorics''' is the study of [[discrete]] structures broadly speaking. Most notably, combinatorics involves studying the enumeration (counting) of said structures. For example, the number of three-[[cycle|cycles]] in a given [[graph]] is a combinatorial problem, as is the derivation of a non-[[recursive]] formula for the [[Fibonacci numbers]], and so too methods of solving the [[Rubik's cube]]. Mathematicians who spend their careers studying combinatorics are known as ''combinatorialists''. |
| − | + | Combinatorial problems often make up a good portion of problems found in mathematics competitions and can be approached by a variety of techniques, such as [[generating functions]] or the [[Principle of Inclusion-Exclusion|principle of inclusion-exclusion]]. Combinatorics also has many applications outside of pure mathematics, notably in [[theoretical computer science]], [[statistics]], and various fields of science. | |
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| − | + | People who encounter the term combinatorics for the first time often discredit it as "easy" because they "already know how to count." While this is true in the sense that people know how to count lists of numbers, enumeration problems are (typically) not nearly as simple as counting a list of numbers. One must first determine ''what'' and ''how'' something must be counted, both of which are often difficult to do. | |
| − | + | == Study Guides to Combinatorics == | |
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| − | + | * [[Combinatorics/Introduction]] | |
| + | * [[Combinatorics/Intermediate]] | ||
| + | * [[Combinatorics/Olympiad]] | ||
| − | + | == Resources == | |
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Listed below are various combinatorics resources including books, classes, and websites. | Listed below are various combinatorics resources including books, classes, and websites. | ||
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* Introductory | * Introductory | ||
| − | ** ''the Art of Problem Solving Introduction to Counting and Probability'' by David Patrick [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php? | + | ** ''the Art of Problem Solving Introduction to Counting and Probability'' by David Patrick [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=202 (details)] |
| + | * Intermediate | ||
| + | ** ''the Art of Problem Solving Intermediate Counting and Probability'' by David Patrick [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=302 (details)] | ||
| + | **''Combinatorics:A Guided Tour'' by David R. Mazur.Follow this [//books.google.co.in/books?id=yI4Jx5Obr08C&pg=PA15&lpg=PA15&dq=overcounting+combinatorics&source=bl&ots=5R05q-YhEq&sig=xLGDWSWSn52t1h2ZVmbzqQ8qhHw&hl=en&sa=X&ei=OwKyT9_2C4qo2wXV1bj-CA&sqi=2#v=onepage&q=overcounting%20combinatorics&f=false (link)] | ||
| + | * Undergraduate | ||
| + | ** [//www.math.upenn.edu/~wilf/DownldGF.html ''Generatingfunctionology'' by Herbert S. Wilf] | ||
| − | == See | + | == See Also == |
* [[Probability]] | * [[Probability]] | ||
| + | * [[:Category:Introductory Combinatorics Problems|Introductory Problems]] | ||
| + | * [[:Category:Intermediate Combinatorics Problems|Intermediate Problems]] | ||
| + | * [[:Category:Olympiad Combinatorics Problems|Olympiad Problems]] | ||
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| + | [[Category:Combinatorics]] | ||
| + | [[Category:Mathematics]] | ||
| + | {{stub}} | ||
Latest revision as of 19:02, 11 March 2025
Combinatorics is the study of discrete structures broadly speaking. Most notably, combinatorics involves studying the enumeration (counting) of said structures. For example, the number of three-cycles in a given graph is a combinatorial problem, as is the derivation of a non-recursive formula for the Fibonacci numbers, and so too methods of solving the Rubik's cube. Mathematicians who spend their careers studying combinatorics are known as combinatorialists.
Combinatorial problems often make up a good portion of problems found in mathematics competitions and can be approached by a variety of techniques, such as generating functions or the principle of inclusion-exclusion. Combinatorics also has many applications outside of pure mathematics, notably in theoretical computer science, statistics, and various fields of science.
People who encounter the term combinatorics for the first time often discredit it as "easy" because they "already know how to count." While this is true in the sense that people know how to count lists of numbers, enumeration problems are (typically) not nearly as simple as counting a list of numbers. One must first determine what and how something must be counted, both of which are often difficult to do.
Study Guides to Combinatorics
Resources
Listed below are various combinatorics resources including books, classes, and websites.
Books
- Introductory
- the Art of Problem Solving Introduction to Counting and Probability by David Patrick (details)
- Intermediate
- Undergraduate
See Also
This article is a stub. Help us out by expanding it.
