Combinatorics is an area of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of

pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...

, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the

analysis of algorithms In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that re ...

. A mathematician who studies combinatorics is called a '.

Definition

The full scope of combinatorics is not universally agreed upon. According to

H.J. Ryser Herbert John Ryser (July 28, 1923 – July 12, 1985) was a professor of mathematics, widely regarded as one of the major figures in combinatorics in the 20th century.Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained." One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella. Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.

History

Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE,

ancient Indian The following outline is provided as an overview of and topical guide to ancient India: Ancient India is the Indian subcontinent from prehistoric times to the start of Medieval India, which is typically dated (when the term is still used) to th ...

physician

Sushruta Sushruta, or ''Suśruta'' (Sanskrit: सुश्रुत, IAST: , ) was an ancient Indian physician. The ''Sushruta Samhita'' (''Sushruta's Compendium''), a treatise ascribed to him, is one of the most important surviving ancient treatises on ...

asserts in

Sushruta Samhita The ''Sushruta Samhita'' (सुश्रुतसंहिता, IAST: ''Suśrutasaṃhitā'', literally "Suśruta's Compendium") is an ancient Sanskrit text on medicine and surgery, and one of the most important such treatises on this subj ...

that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 2⁶ − 1 possibilities. Greek historian Plutarch discusses an argument between

Chrysippus Chrysippus of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of the Stoic philosopher Cleanthes. When Clean ...

(3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers. Earlier, in the '' Ostomachion'',

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

(3rd century BCE) may have considered the number of configurations of a tiling puzzle, while combinatorial interests possibly were present in lost works by Apollonius. In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra () provided formulae for the number of

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s and combinations, and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE. The

philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

and astronomer Rabbi Abraham ibn Ezra () established the symmetry of

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321. The arithmetical triangle—a graphical diagram showing relationships among the binomial coefficients—was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of

Pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Fren ...

Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...

, Jacob Bernoulli and

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

became foundational in the emerging field. In modern times, the works of

J.J. Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership r ...

(late 19th century) and

Percy MacMahon Percy Alexander MacMahon (26 September 1854 – 25 December 1929) was a mathematician, especially noted in connection with the partitions of numbers and enumerative combinatorics. Early life Percy MacMahon was born in Malta to a British mi ...

(early 20th century) helped lay the foundation for

enumerative An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (f ...

and

algebraic combinatorics Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algeb ...

. Graph theory also enjoyed an increase of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject. In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field.

Approaches and subfields of combinatorics

Enumerative combinatorics

Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad

mathematical problem A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...

, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.

Analytic combinatorics

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.

Partition theory

Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...

and has connections with

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of

and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.

Graph theory

Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on ''n'' vertices with ''k'' edges) to existing structures (e.g., Hamiltonian cycles) to algebraic representations (e.g., given a graph ''G'' and two numbers ''x'' and ''y'', does the Tutte polynomial ''T''_''G''(''x'',''y'') have a combinatorial interpretation?). Although there are very strong connections between graph theory and combinatorics, they are sometimes thought of as separate subjects. While combinatorial methods apply to many graph theory problems, the two disciplines are generally used to seek solutions to different types of problems.

Design theory

Design theory is a study of combinatorial designs, which are collections of subsets with certain

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

properties.

Block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...

s are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics. Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry,

mathematical biology Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...

, algorithm design and analysis,

networking Network, networking and networked may refer to: Science and technology * Network theory, the study of graphs as a representation of relations between discrete objects * Network science, an academic field that studies complex networks Mathematics ...

, group testing and cryptography.

Finite geometry

Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory. It should not be confused with discrete geometry ( combinatorial geometry).

Order theory

Order theory is the study of partially ordered sets, both finite and infinite. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". Various examples of partial orders appear in algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and

Boolean algebras In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...

Matroid theory

Matroid theory 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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory. It is now an independent field of study with a number of connections with other parts of combinatorics.

Extremal combinatorics

Extremal combinatorics studies how large or how small a collection of finite objects ( numbers,

graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of set systems; this is called extremal set theory. For instance, in an ''n''-element set, what is the largest number of ''k''-element

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by

Sperner's theorem Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory. It is named after Emanuel Sperner, who ...

, which gave rise to much of extremal set theory. The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on ''2n'' vertices is a complete bipartite graph ''K_n,n''. Often it is too hard even to find the extremal answer ''f''(''n'') exactly and one can only give an

asymptotic estimate In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...

. Ramsey theory is another part of extremal combinatorics. It states that any

sufficiently large In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have pas ...

configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle.

Probabilistic combinatorics

In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find) by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as ''the'' probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time. Often associated with

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. However, with the growth of applications to analyze algorithms in computer science, as well as classical probability, additive number theory, and probabilistic number theory, the area recently grew to become an independent field of combinatorics.

Algebraic combinatorics

Algebraic combinatorics is an area of

that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be

in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.

Combinatorics on words

Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory, and linguistics. While many applications are new, the classical

Chomsky–Schützenberger hierarchy In formal language theory, computer science and linguistics, the Chomsky hierarchy (also referred to as the Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars. This hierarchy of grammars was described by ...

of classes of formal grammars is perhaps the best-known result in the field.

Geometric combinatorics

Geometric combinatorics is related to convex and discrete geometry. It asks, for example, how many faces of each dimension a

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. Combinatorial geometry is a historical name for discrete geometry. It includes a number of subareas such as

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral comb ...

(the study of faces of

convex polyhedra A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to

computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

. The study of regular polytopes,

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s, and kissing numbers is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron, associahedron and Birkhoff polytope.

Topological combinatorics

Combinatorial analogs of concepts and methods in topology are used to study

graph coloring In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...

, fair division, partitions, partially ordered sets,

decision tree A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains condit ...

s, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.

Arithmetic combinatorics

Arithmetic combinatorics arose out of the interplay between number theory, combinatorics,

ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

, and

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the

of dynamical systems.

Infinitary combinatorics

Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the

continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...

and combinatorics on successors of singular cardinals. Gian-Carlo Rota used the name ''continuous combinatorics'' to describe geometric probability, since there are many analogies between ''counting'' and ''measure''.

Related fields

Combinatorial optimization

Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory.

Coding theory

Coding theory started as a part of design theory with early combinatorial constructions of

error-correcting code In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is ...

s. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of

information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

Discrete and computational geometry

Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on

s and kissing numbers. With the emergence of applications of discrete geometry to

, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.

Combinatorics and dynamical systems

Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system.

Combinatorics and physics

There are increasing interactions between combinatorics and physics, particularly statistical physics. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the

chromatic Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, ...

and Tutte polynomials on the other hand.

Notes

References

* Björner, Anders; and Stanley, Richard P.; (2010)
''A Combinatorial Miscellany''
* Bóna, Miklós; (2011)
''A Walk Through Combinatorics (3rd Edition)''
* Graham, Ronald L.; Groetschel, Martin; and Lovász, László; eds. (1996); ''Handbook of Combinatorics'', Volumes 1 and 2. Amsterdam, NL, and Cambridge, MA: Elsevier (North-Holland) and MIT Press. * Lindner, Charles C.; and Rodger, Christopher A.; eds. (1997); ''Design Theory'', CRC-Press; 1st. edition (1997). . * * * Stanley, Richard P. (1997, 1999)
''Enumerative Combinatorics'', Volumes 1 and 2
Cambridge University Press. * * van Lint, Jacobus H.; and Wilson, Richard M.; (2001); ''A Course in Combinatorics'', 2nd Edition, Cambridge University Press.

External links

*
Combinatorial Analysis
– an article in Encyclopædia Britannica Eleventh Edition
Combinatorics
a MathWorld article with many references.
Combinatorics
from a ''MathPages.com'' portal.
The Hyperbook of Combinatorics
a collection of math articles links.
The Two Cultures of Mathematics
by W.T. Gowers, article on problem solving vs theory building.

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