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). Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Combinatorial enumeration is a subfield of enumeration that deals with the counting of objects whose symmetries do not exist or, if they exist, are combinatorial in nature. ... Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties. ... In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence (hence independence structure) that generalizes linear independence in vector spaces. ... Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. ... Combinatorial optimization is a branch of optimization in applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory. ... Algebra (from Arabic: الجبر, al-ğabr) is a branch of mathematics concerning the study of structure, relation and quantity. ...

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. A labeled graph with 6 vertices and 7 edges. ...

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. In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ... Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. ... Avogadros number, also called Avogadros constant (NA) is a large constant used in chemistry and physics. ...

An example of another kind is this problem: Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains all or all but one of the people? The answer depends on n. See "Design theory" below.

Overview and history

The earliest recorded statements of combinatorical rules appear in India. The medical treatise Sushruta Samhita written by Sushruta in the 6th century BC states that 63 combinations can be made out of six different tastes – bitter, sour, salty, sweet, astringent, and hot – by taking them one at a time, two at a time, three at a time, etc. In other words, there are 6 single tastes, 15 combinations of two, 20 combinations of three, etc. The Bhagabati Sutra, written by a Jaina mathematician circa 300 BC, contains rules on combinations and permutations corresponding to: This article is about the field of medical practice and health care. ... Sushruta (also spelt Susruta or Sushrutha) (c. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 6th century BC started on January 1, 600 BC and ended on December 31, 501 BC. // Overview Monument 1, an Olmec colossal head at La Venta The 5th and 6th centuries BC were a time of empires, but more importantly, a... It has been suggested that Permutations and combinations be merged into this article or section. ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC... This article is an elementary introduction to permutations and combinations in combinatorial mathematics. ...

and:

Numbers are calculated in the cases where n = 2, 3 and 4. The author then says that one can compute the numbers in the same way for larger n: "In this way, 5, 6, 7, ..., 10, etc. or an enumerable, unenumerable or infinite number of things may be specified. Taking one at a time, two at a time, ... ten at a time, as the number of combinations are formed they must all be worked out." This suggests that the arithmetic can be extended to various infinite numbers. The relation of the number of combinations to the coefficients occurring in the binomial expansion was noted by Pingala in the 3rd century BC in a musical composition. He gave the different combinations of guru and laghu sounds as a meru-prastara (Pascal's triangle) and gave a rule simpler than that of Blaise Pascal, based on the simple formula Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 3rd century BC started on January 1, 300 BC and ended on December 31, 201 BC. // Events The Pyramid of the Moon, one of several monuments built in Teotihuacán Teotihuacán, Mexico begun The first two Punic Wars between Carthage... 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... Blaise Pascal (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ...

Varahamihira in the 6th century CE states that "if a quantity of 16 substances is varied in four different ways, the result will be 1820." He found this result using rules related to Pascal's triangle. In the 9th century, Mahavira gave an explicit algorithm for calculating the number of combinations and provided the well-known general formula Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ... This Buddhist stela from China, Northern Wei period, was built in the early 6th century. ... As a means of recording the passage of time the 9th century was that century that lasted from 801 to 900. ... Mahavira, 9th century Indian mathematician, asserted that the square root of a negative number did not exist. ...

Muslim mathematicians later studied combinatorial analysis from at least the 13th century. Ibn Mun'im, in the Maghreb of North Africa in the early 13th century, dealt with combinatory problems. He stated the rule for determining all possible combinations of n colours p times and established, inductively, the resulting arithmetic triangle of the relationship as Islamic mathematics is the profession of Muslim Mathematicians. ... (12th century - 13th century - 14th century - other centuries) As a means of recording the passage of time, the 13th century was that century which lasted from 1201 to 1300. ... Marrakesh, Morocco, in front of Atlas Mountains in Maghreb The Maghreb (المغرب العربي ; also rendered Maghrib (or rarely Moghreb), meaning western in Arabic, is the region of Africa north of the Sahara Desert and west of the Nile — specifically, coinciding with the Atlas Mountains. ...  Northern Africa (UN subregion)  geographic, including above North Africa or Northern Africa is the northernmost region of the African continent. ...

He applied similar formulae for permutations with and without repetitions using the Arabic alphabet for illustrative purposes. He also does some work on combinatorial reasoning. Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ...

Persian mathematician Al-Farisi, in the late 13th century, introduced ideas concerning factorisation and combinatorial methods. Al-Farisi's approach is based on the unique factorisation of an integer into powers of prime numbers. He states and proves this fundamental theorem of arithmetic. Al-Farisi saw the relation between polygonal numbers and the binomial coefficients and he presented arguments, using an early form of mathematical induction, which showed a relation between triangular numbers, the sums of triangular numbers, the sums of the sums of triangular number, etc., and the combinations of n objects taken k at a time. The Persians are an Iranian people who speak the Persian language and share a common culture and history. ... Kamal al-Din Abul Hasan Muhammad Al-Farisi (in Persian: كمال الدين ابوالحسن محمد الفارسي) (1260 - 1320) was a prominent Persian mathematician and physicist. ... (12th century - 13th century - 14th century - other centuries) As a means of recording the passage of time, the 13th century was that century which lasted from 1201 to 1300. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

Enumerative combinatorics came to prominence in Europe after counting configurations became essential to elementary probability, starting with the work of Pascal and others from the 17th century. Modern combinatorics began developing in the late 19th century and became a distinguishable field of study in the 20th century, partly through the publication of the systematic enumerative treatise Combinatory Analysis by Percy Alexander MacMahon in 1915 and the work of R.A. Fisher in design of experiments in the 1920s. Two of the most prominent combinatorialists of recent times were the prolific problem-raiser and problem-solver Paul Erdős, who worked mainly on extremal questions, and Gian-Carlo Rota, who helped to formalize the subject beginning in the 1960s, mostly in enumeration and algebraization. The study of how to count objects is sometimes thought of separately as the field of enumeration. The word probability derives from the Latin probare (to prove, or to test). ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... Percy Alexander MacMahon (b. ... 1915 (MCMXV) was a common year starting on Friday (see link for calendar). ... Sir Ronald Fisher Sir Ronald Aylmer Fisher, FRS (February 17, 1890–July 29, 1962) was an extraordinarily talented evolutionary biologist, geneticist and statistician. ... The first statistician to consider a methodology for the design of experiments was Sir Ronald A. Fisher. ... It has been suggested that this article or section be merged with Social issues of the 1920s. ... Paul ErdÅ‘s also Pál ErdÅ‘s, in English Paul Erdos or Paul Erdös, (March 26, 1913 – September 20, 1996) was an immensely prolific (and famously eccentric) Hungarian mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set... Gian-Carlo Rota (April 27, 1932 – April 18, 1999, known as Juan Carlos Rota to Spanish speakers) was an Italian-born American mathematician and philosopher. ... The 1960s decade refers to the years from 1960 to 1969, inclusive. ... suvodip ...

Permutations and combinations

Permutation with repetition

When order matters and an object can be chosen more than once then the number of permutations is

where n is the number of objects from which you can choose and r is the number to be chosen.

For example, if you have the letters A, B, C, and D and you wish to discover the number of ways to arrange them in three letter patterns (trigrams) you find that there are 4³ or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total. The bagua (Chinese: 八卦; pinyin: ; Wade-Giles: pa kua; literally eight trigrams) is a fundamental philosophical concept in ancient China. ...

Permutation without repetition

When the order matters and each object can be chosen only once, then the number of permutations is

where n is the number of objects from which you can choose, r is the number to be chosen and "!" is the standard symbol meaning factorial. In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ...

For example, if you have five people and are going to choose three out of these, you will have 5!/(5−3)! = 60 permutations.

Note that if n = r (meaning number of chosen elements is equal to number of elements to choose from) then the formula becomes

when 0! = 1! = 1.

For example, if you have three people and you want to find out how many ways you may arrange them it would be 3! or 3 × 2 × 1 = 6 ways. The reason for this is because you can choose from 3 for the initial slot, then you are left with only two to choose from for the second slot, and that leaves only one for the final slot. Multiplying them together gives the total.

Combination without repetition

When the order does not matter, but each object can be chosen only once, the number of combinations is the binomial coefficient In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...

where n is the number of objects from which you can choose and r is the number to be chosen.

For example, if you have ten numbers and wish to choose 5 you would have 10!/(5!(10−5)!) = 252 ways to choose.

Combination with repetition

When the order does not matter and an object can be chosen more than once, then the number of combinations is

where n is the number of objects from which you can choose and r is the number to be chosen.

For example, if you have ten types of donuts to choose from and you want three donuts there are (10 + 3 − 1)! / 3!(10 − 1)! = 220 ways to choose (see also multiset). In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ...

Enumerative combinatorics

Calculating the number of ways that certain patterns can be formed is the beginning of combinatorics. Let S be a set with n objects. Combinations of k objects from this set S are subsets of S having k elements each (where the order of listing the elements does not distinguish two subsets). Permutations of k objects from this set S refer to sequences of k different elements of S (where two sequences are considered different if they contain the same elements but in a different order, or if they have a different length). Formulas for the number of permutations and combinations are readily available and important throughout combinatorics. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... It has been suggested that Permutations and combinations be merged into this article or section. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... This article is an elementary introduction to permutations and combinations in combinatorial mathematics. ...

More generally, given an infinite collection of finite sets {S_i} typically indexed by the natural numbers, enumerative combinatorics seeks a variety of ways of describing a counting function, f(n), which counts the number of objects in S_n for any n. Although the activity of counting the number of elements in a set is a rather broad mathematical problem, in a combinatorial problem the elements S_i will usually have a relatively simple combinatorial description, and little additional structure. In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... A mathematical problem is a problem that can be solved with the help of mathematics. ...

The simplest such functions are closed formulas, which can be expressed as a composition of simple functions like factorials, powers, and so on. For instance, and as noted above, the number of possible different orderings of a deck of n cards is f(n) = n!. In mathematics, an equation or system of equations is said to have a closed-form solution just in case a solution can be expressed analytically in terms of a bounded number of well_known operations. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ...

This approach may not always be entirely satisfactory (or practical). For example, let f(n) be the number of distinct subsets of the integers in the interval [1,n] that do not contain two consecutive integers; e.g., with n = 4, we have the sets {}, {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,4}, so f(4) = 8. It turns out that f(n) is the n+2nd Fibonacci number, F(n+2), so it can be expressed in closed form as The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the Fibonacci numbers, named after Leonardo of Pisa, known as Fibonacci, form a sequence defined recursively by: In other words, after two starting values, each number is the sum of the two numbers before it. ...

where , the golden ratio. However, given that we are looking at a counting function, the presence of the in the result may be considered unaesthetic. As an alternative that shows more clearly why f(n) is a positive integer, f(n) may be expressed by the recurrence relation In many branches of mathematics, the golden ratio is the irrational number given by The golden ratio is also known by many other names such as the golden proportion, golden mean, golden section, golden number, divine proportion, φ, sectio divina, or mean and extreme ratio. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

with the initial conditions f(1) = 1 and f(2) = 1.

Another approach is to find an asymptotic formula In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

where g(n) is a "familiar" function, and where f(n) approaches g(n) as n approaches infinity. In some cases, a simple asymptotic function may be preferable to a horribly complicated closed formula that yields no insight to the behaviour of the counted objects. In the above example, an asymptotic formula would be The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ...

as n becomes large.

Finally, f(n) may be expressed by a formal power series, called its generating function, which is most commonly either the ordinary generating function In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

or the exponential generating function In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

Once determined, the generating function may allow one to extract all the information given by the previous approaches. In addition, the various natural operations on generating functions such as addition, multiplication, differentiation, etc., have a combinatorial significance; this allows one to extend results from one combinatorial problem in order to solve others.

Structural combinatorics

There are many combinatorial patterns and theorems related to the structure of combinatoric sets. These often focus on a partition or ordered partition of a set. See the List of partition topics for an expanded list of related topics or the List of combinatorics topics for a more general listing. Some of the more notable results are highlighted below. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... A partition of U into 6 blocks: a Venn diagram representation. ... In combinatorial mathematics, an ordered partition O of a set S is a sequence A1, A2, A3, ..., An of subsets of S, with union is S, which are non-empty, and pairwise disjoint. ... This is a list of partition topics, in the mathematical sense. ... This is a list of combinatorics topics, by Wikipedia page. ...

Design theory

A simple result in this area of combinatorics is that the problem of forming sets, described in the introduction, has a solution only if n has the form q² + q + 1. It is less simple to prove that a solution exists if q is a prime power. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for q congruent to 1 or 2 mod 4, then q is a sum of two square numbers. This last result, the Bruck-Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms. You may be looking for block design test In combinatorial mathematics, a block design is a particular kind of set system, which has some long-standing applications to experimental design, as well as some pure combinatorial aspects. ... A prime power is a positive integer power of a prime. ... In mathematics, a square number, sometimes also called a perfect square, is a positive integer that can be written as the square of some other integer. ... The Bruck–Chowla–Ryser theorem is a result on the combinatorics of block designs. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combinatorics intersect. Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... A finite geometry is any geometric system that has only a finite number of points. ...

Ramsey theory

Frank P. Ramsey proved that, given any group of six people, it is always the case that one can find three people out of this group that either all know each other, or all do not know each other. Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ... Frank Plumpton Ramsey (February 22, 1903 – January 19, 1930) was a British mathematician, philosopher, and economist. ...

The proof is a short proof by contradiction: suppose the claim is false. This means that we can have a group of six people such that whenever we look at any three of the six, there are at least two people among these three that know each other and at least two who do not know each other. Consider now one person among the six; call this person "A." Now, among the remaining five people, there must be at least three who either all know A or all do not know A--this is clear since the negation of one condition immediately implies the other condition. Assume first former condition: that at least three of the remaining five know A. Among those three people, at least two of them must know each other, since otherwise we would have three people who all don't know each other, contrary to our hypothesis. But then we have two people who know each other, and know A, and so these two people, along with A, constitute a group of three people among the six who all know each other. This contradicts our initial hypothesis. Assuming that other condition--that three of the remaining five do not know A--results in a similar contradiction. Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then...

This is a special case of Ramsey's theorem. In combinatorics, Ramseys theorem states that in colouring a large complete graph (that is a simple graph, where an edge connects every pair of vertices), one will find complete subgraphs all of the same colour. ...

The idea of finding order in random configurations gives rise to Ramsey theory. Essentially this theory says that any sufficiently large configuration will contain at least one instance of some other type of configuration. Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ... In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ...

Matroid theory

This part of combinatorics abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence that generalizes linear independence in vector spaces. ... Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ...

For instance, given a set of n vectors in Euclidean space, what is the largest number of planes they can generate? (Answer: the binomial coefficient C(n,3).) Is there a set that generates exactly one less plane? (No, in almost all cases.) These are extremal questions in geometry. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...

Extremal combinatorics

Many extremal questions deal with set systems. A simple example is the following: what is the largest number of subsets of an n-element set one can have, if no two of the subsets are disjoint? Answer: half the total number of subsets. Proof: Call the n-element set S. Between any subset T and its complement S − T, at most one can be chosen. This proves the maximum number of chosen subsets is not greater than half the number of subsets. To show one can attain half the number, pick one element x of S and choose all the subsets that contain x. In mathematics, the concept of hypergraph generalizes the notion of a graph. ... The word complement (with an e in the second syllable, not to be confused with a different word, compliment with an i) has a number of uses. ...

A more difficult problem is to characterize the extremal solutions; in this case, to show that no other choice of subsets can attain the maximum number while satisfying the requirement.

Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate. In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

See also

Wikimedia Commons has media related to:

Category:Combinatorics

• Permutations and combinations
• Combinatorial principles
• Inclusion-exclusion principle
• Method of distinguished element
• Important publications in combinatorics
• List of combinatorics topics
• Musical set theory
• Combinatorial chemistry
• Combinatorial explosion

Image File history File links Commons-logo. ... Wikimedia Commons logo by Reid Beels The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ... This article is an elementary introduction to permutations and combinations in combinatorial mathematics. ... In proving results in combinatorics several useful combinatorial rules or combinatorial principles are used. ... In combinatorial mathematics, the inclusion-exclusion principle states that if A1, ..., An are finite sets, then where |A| denotes the cardinality of the set A. For example, taking n = 2, we get a special case of double counting: in words, we can count the size of the union of sets... In enumerative combinatorial mathematics, identities are sometimes established by arguments that rely on singling out one distinguished element of a set. ... This is a list of important publications in mathematics, organized by field. ... This is a list of combinatorics topics, by Wikipedia page. ... Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ... Combinatorial chemistry involves the rapid synthesis and/or the computer simulation of a large number of different but structurally related molecules. ... In cryptanalysis, a brute force attack on a cipher is a brute-force search of the key space; that is, testing all possible keys, in an attempt to recover the plaintext used to produce a particular ciphertext. ...

References

• Graham, R.L., Groetschel M., and Lovász L., eds. (1996). Handbook of Combinatorics, Volumes 1 and 2. Elsevier (North-Holland), Amsterdam, and MIT Press, Cambridge, Mass. ISBN 0-26207169-X.
• Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books. ISBN 0-14-021112-1.
• Katz, Victor J. (1998). A History of Mathematics: An Introduction, 2nd Edition. Addison-Wesley Education Publishers. ISBN 0-32101618-1.
• Lindner, Charles C. and Christopher A. Rodger (eds.) Design Theory, CRC-Press; 1st. edition (October 31, 1997). ISBN 0849339863.
• van Lint, J.H., and Wilson, R.M. (2001). A Course in Combinatorics, 2nd Edition. Cambridge University Press. ISBN 0-521-80340-3.
• O'Connor, John J. and Robertson, Edmund F. (1999-2004). MacTutor History of Mathematics archive. St Andrews University.
• Rashed, R. (1994). The development of Arabic mathematics : between arithmetic and algebra. London.
• Stanley, Richard P. (1997, 1999). Enumerative Combinatorics, Volumes 1 and 2. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1.