Combinatorics is an area of

^{6} − 1 possibilities.

_{''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.

_{n,n}''. Often it is too hard even to find the extremal answer ''f''(''n'') exactly and one can only give an asymptotic estimate.

number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, combinatorics, ergodic theory, and harmonic analysis. 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 ergodic theory of dynamical systems.

''Continuous and profinite combinatorics''

/ref> to describe geometric probability, since there are many analogies between ''counting'' and ''measure''.

''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). . * * * Richard P. Stanley, 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.

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.

{{Authority control Combinatorics,

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

structures
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A sy ...

. It is closely related to many other areas of mathematics and has many applications ranging from logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

to statistical physics
Statistical physics is a branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...

and from evolutionary biology
Evolutionary biology is the subfield of biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interacti ...

to computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

.
The full scope of combinatorics is not universally agreed upon. According to H.J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with:
* the ''enumeration'' (counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems,
* the ''existence'' of such structures that satisfy certain given criteria,
* the ''construction'' of these structures, perhaps in many ways, and
* ''optimization'': finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other ''optimality criterion''.
Leon Mirsky
Leonid Mirsky (19 December 1918 Russia
Russia (russian: link=no, Россия, , ), or the Russian Federation, is a country spanning Eastern Europe and Northern Asia. It is the List of countries and dependencies by area, largest country in ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) but discrete
Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual.
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic c ...

setting.
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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

, notably in algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...

, topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, and geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

, 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
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations ''N'' versus input size ''n'' for each function
In computer science
Computer science deals with the theoretical foundations of information, al ...

.
A mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

who studies combinatorics is called a '.
History

Basic combinatorial concepts and enumerative results appeared throughout theancient world
Ancient history is the aggregate of past eventsWordNet Search – 3.0

"History" from t ...

. In the 6th century BCE, ancient Indian "History" from t ...

physician
A physician (American English), medical practitioner (English in the Commonwealth of Nations, Commonwealth English), medical doctor, or simply doctor, is a professional who practices medicine, which is concerned with promoting, maintainin ...

Sushruta
Sushruta, or ''Suśruta'' (Sanskrit
Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-Eu ...

asserts in Sushruta Samhita
The ''Sushruta Samhita'' (सुश्रुतसंहिता, IAST
The International Alphabet of Sanskrit Transliteration (IAST) is a transliteration scheme that allows the lossless romanisation of Brahmic family, Indic scripts as employ ...

that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 2Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...

historian
A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human race; as well as the stu ...

Plutarch
Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; AD 46 – after AD 119) was a Greek Middle Platonist
Middle Platonism is the modern name given to a stage in the development of Platonic philosophy, lasting from about 90 BC&nbs ...

discusses an argument between Chrysippus
Chrysippus of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) was a Greek Stoic philosopher
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosop ...

(3rd century BCE) and Hipparchus
Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the ...

(2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus number
In combinatorics
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, c ...

s. Earlier, in the ''Ostomachion
{{original research, section, date=October 2013}
''Ostomachion'', also known as ''loculus Archimedius'' (Archimedes' box in Latin language, Latin) and also as ''syntomachion'', is a mathematical treatise attributed to Archimedes. This work has ...

'', Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

(3rd century BCE) may have considered the number of configurations of a tiling puzzle
Tiling puzzles are puzzle
A puzzle is a game
A game is a structured form of play
Play most commonly refers to:
* Play (activity), an activity done for enjoyment
* Play (theatre), a work of drama
Play may refer also to:
Computers and ...

, while combinatorial interests possibly were present in lost works by Apollonius.
In the Middle Ages
In the history of Europe
The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...

, combinatorics continued to be studied, largely outside of the European civilization. The India
India, officially the Republic of India (Hindi
Hindi (Devanagari: , हिंदी, ISO 15919, ISO: ), or more precisely Modern Standard Hindi (Devanagari: , ISO 15919, ISO: ), is an Indo-Aryan language spoken chiefly in Hindi Belt, ...

n mathematician Mahāvīra
Mahavira (Sanskrit
Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. ...

(c. 850) provided formulae for the number of permutation
In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s and combination
In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of t ...

s, and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE. The philosopher
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ...

and astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...

Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s, while a closed formula was obtained later by the talmudist
The Talmud (; he, תַּלְמוּד ''Tálmūḏ'') is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the ...

and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

Levi ben Gerson
Levi ben Gershon (1288 – 1344), better known by his Graecized name as Gersonides, or by his Latinized name Magister Leo Hebraeus, or in Hebrew
Hebrew (, , or ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasia ...

(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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

. Later, in Medieval England
England in the Middle Ages concerns the history of England
The British Isles became inhabited more than 800,000 years ago, as the discovery of stone tools and footprints at Happisburgh in Norfolk has indicated.; "Earliest footprints outside ...

, campanology
Campanology (from Late Latin
Late Latin ( la, Latinitas serior) is the scholarly name for the written Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was ori ...

provided examples of what is now known as Hamiltonian cycle
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s in certain Cayley graph on two generators ''a'' and ''b''
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

s on permutations.
During the Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in ...

, together with the rest of mathematics and the science
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is something that is truth, true. The usual test for a statement of ...

s, 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, French ...

, 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), ''Newton'' (film), a 2017 Indian fil ...

, Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) inclu ...

and Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

became foundational in the emerging field. In modern times, the works of (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics
Algebraic combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...

. Graph theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

also enjoyed an increase of interest at the same time, especially in connection with the four color problem
In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into wikt:contiguity, contiguous regions, producing a figure called a ''map'', no more than four colors are required to color the r ...

.
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
Functional analysis is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...

to number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, 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 broadmathematical problem
A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ...

, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
F_0=0,\quad F_1= 1,
and
F_n=F_ + F_
for .
Th ...

is the basic example of a problem in enumerative combinatorics. The twelvefold way
In combinatorics
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, ...

provides a unified framework for counting permutations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, combinations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and .
Analytic combinatorics

Analytic combinatorics
''Analytic Combinatorics'' is a book on the mathematics of combinatorial enumeration, using generating functions and complex analysis to understand the growth rates of the numbers of combinatorial objects. It was written by Philippe Flajolet and Rob ...

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
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Der ...

and probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...

. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.
Partition theory

Partition theory studies various enumeration and asymptotic problems related tointeger partition
300px, Partitions of ''n'' with biggest addend ''k''
In number theory and combinatorics
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mat ...

s, and is closely related to q-series
In mathematics, in the area of combinatorics, a ''q''-Pochhammer symbol, also called a ''q''-shifted factorial, is a q-analog, ''q''-analog of the Pochhammer symbol. It is defined as
:(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^)
w ...

, special functions
Special functions are particular function (mathematics), mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. ...

and orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomials ...

. Originally a part of number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

and analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

and has connections with statistical mechanics
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

.
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 theTutte polynomial
. The red line shows the intersection with the plane y=0, equivalent to the chromatic polynomial.
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial
In mathematics
...

''T''Design theory

Design theory is a study ofcombinatorial design
Combinatorial design theory is the part of combinatorial mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and c ...

s, which are collections of subsets with certain intersection
The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.
In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

properties. Block design
In combinatorial mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, ...

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
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in ''The Lady's and Gentleman's Diary'' (pg.48). The problem states:
Fifteen young ladies in a school walk out three abreast ...

proposed in 1850. The solution of the problem is a special case of a Steiner system 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In Combinatorics, combinatorial mathematics, a Steiner system (named after Jak ...

, which systems play an important role in the classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

. The area has further connections to coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...

and geometric combinatorics.
Finite geometry

Finite geometry is the study of having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

, real projective spaceIn mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n' ...

, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory
Design theory is a subfield of design research
Design research was originally constituted as primarily research into the process of design, developing from work in design methodsDesign methods are procedures, techniques, aids, or tools for desi ...

. It should not be confused with discrete geometry (combinatorial geometry
Image:Unit disk graph.svg, A collection of circles and the corresponding unit disk graph
Discrete geometry and combinatorial geometry are branches of geometry that study Combinatorics, combinatorial properties and constructive methods of discrete ...

).
Order theory

Order theory is the study ofpartially ordered sets
250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable.
In mathematics, especially order the ...

, both finite and infinite. Various examples of partial orders appear in algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, 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
Logic (from Ancient Greek, Greek: grc, wi ...

.
Matroid theory

Matroid theory abstracts part ofgeometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

. It studies the properties of sets (usually, finite sets) of vectors in a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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 at least one of the vectors in the set can be defined as a linear combinationIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersion (mathematics), immersions, characteristic classes, and ...

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 extremal questions onset system In set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of ...

s. The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graphIn the mathematical area of graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up ...

on ''2n'' vertices is a complete bipartite graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex (graph theory), vertex of the first set is connected to every vertex of the second set..Electronic edition pa ...

''KRamsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related s ...

is another part of extremal combinatorics. It states that any sufficiently large
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population i ...

configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
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 arandom graph
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

? 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), simply 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
The probabilistic method is a nonconstructive method, primarily used in combinatorics
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...

) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains
A Markov chain is a stochastic model
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in ...

, 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 renowned Hungarian mathematician
In this page we keep the names in Hungarian order (family name first).
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* Alexits György (1899–1 ...

, 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
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

, as well as classical probability, additive number theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigro ...

, and probabilistic number theory, the area recently grew to become an independent field of combinatorics.
Algebraic combinatorics

Algebraic combinatorics is an area ofmathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

that employs methods of abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, notably group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

and representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, and can be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant.
Combinatorics on words

Combinatorics on words deals withformal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...

s. It arose independently within several branches of mathematics, including number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

and probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

. It has applications to enumerative combinatorics, fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from ph ...

, theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ...

, automata theory
Automata theory is the study of abstract machines and automaton, automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' (the plural of ''automaton'') com ...

, and linguistics
Linguistics is the scientific study of language
A language is a structured system of communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo ...

. While many applications are new, the classical Chomsky–Schützenberger hierarchy of classes of formal grammars is perhaps the best-known result in the field.
Geometric combinatorics

Geometric combinatorics is related to Convex geometry, convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension a convex polytope can have. Metric geometry, Metric properties of polytopes play an important role as well, e.g. the Cauchy's theorem (geometry), Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedron, permutohedra, associahedron, associahedra and Birkhoff polytopes. Combinatorial geometry is a historical name for discrete geometry.Topological combinatorics

Combinatorial analogs of concepts and methods intopology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

are used to study graph coloring, fair division, partition of a set, partitions, partially ordered sets, decision trees, 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 betweenInfinitary 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. Gian-Carlo Rota used the name ''continuous combinatorics''/ref> 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, Analysis of algorithms, algorithm theory and computational complexity theory.Coding theory

Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. 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.Discrete and computational geometry

Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, 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

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, particularlystatistical physics
Statistical physics is a branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...

. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic polynomial, chromatic and Tutte polynomial
. The red line shows the intersection with the plane y=0, equivalent to the chromatic polynomial.
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial
In mathematics
...

s on the other hand.
See also

* Combinatorial biology * Combinatorial chemistry * Combinatorial data analysis * Combinatorial game theory * Combinatorial group theory * List of combinatorics topics * Phylogenetics * Polynomial method in combinatoricsNotes

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). . * * * Richard P. Stanley, 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.

{{Authority control Combinatorics,