History Of Combinatorics
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The mathematical field of
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
was studied to varying degrees in numerous ancient
societies A society () is a group of individuals involved in persistent social interaction or a large social group sharing the same spatial or social territory, typically subject to the same political authority and dominant cultural expectations. ...
. Its study in
Europe Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. It is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, the Mediterranean Sea to the south, and Asia to the east ...
dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced
Arabian The Arabian Peninsula (, , or , , ) or Arabia, is a peninsula in West Asia, situated north-east of Africa on the Arabian plate. At , comparable in size to India, the Arabian Peninsula is the largest peninsula in the world. Geographically, the ...
and Indian ideas to the continent. It has continued to be studied in the
modern era The modern era or the modern period is considered the current historical period of human history. It was originally applied to the history of Europe and Western history for events that came after the Middle Ages, often from around the year 1500 ...
.


Earliest records

The earliest recorded use of combinatorial techniques comes from problem 79 of the
Rhind papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics. It is one of two well-known mathematical papyr ...
, which dates to the 16th century BC. The problem concerns a certain
geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
, and has similarities to Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total. In Greece,
Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...
wrote that
Xenocrates Xenocrates (; ; c. 396/5314/3 BC) of Chalcedon was a Greek philosopher, mathematician, and leader ( scholarch) of the Platonic Academy from 339/8 to 314/3 BC. His teachings followed those of Plato, which he attempted to define more closely, of ...
of
Chalcedon Chalcedon (; ; sometimes transliterated as ) was an ancient maritime town of Bithynia, in Asia Minor, Turkey. It was located almost directly opposite Byzantium, south of Scutari (modern Üsküdar) and it is now a district of the city of Ist ...
(396–314 BC) discovered the number of different
syllables A syllable is a basic unit of organization within a sequence of speech sounds, such as within a word, typically defined by linguists as a ''nucleus'' (most often a vowel) with optional sounds before or after that nucleus (''margins'', which are ...
possible in the
Greek language Greek (, ; , ) is an Indo-European languages, Indo-European language, constituting an independent Hellenic languages, Hellenic branch within the Indo-European language family. It is native to Greece, Cyprus, Italy (in Calabria and Salento), south ...
. This would have been the first attempt on record to solve a difficult problem in
permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...
and
combinations In mathematics, a combination is a selection of items from a set that has distinct members, 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 t ...
. The claim, however, is implausible: this is one of the few mentions of combinatorics in
Greece Greece, officially the Hellenic Republic, is a country in Southeast Europe. Located on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, North Macedonia and Bulgaria to the north, and Turkey to th ...
, and the number they found, 1.002 × 10 12, seems too round to be more than a guess. Later, an argument between
Chrysippus Chrysippus of Soli (; , ; ) was a Ancient Greece, Greek Stoicism, Stoic Philosophy, 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 Cleanthes ...
(3rd century BC) and
Hipparchus Hipparchus (; , ;  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
(2nd century BC) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers, is mentioned. There is also evidence that in the '' Ostomachion'',
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
(3rd century BC) considered the configurations of a tiling puzzle, while some combinatorial interests may have been present in lost works of
Apollonius Apollonius () is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apo ...
. In
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
, the Bhagavati Sutra had the first mention of a combinatorics problem; the problem asked how many possible
combinations In mathematics, a combination is a selection of items from a set that has distinct members, 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 t ...
of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (
sweet Sweetness is a basic taste most commonly perceived when eating foods rich in sugars. Sweet tastes are generally regarded as pleasurable. In addition to sugars like sucrose, many other chemical compounds are sweet, including aldehydes, ketones, ...
,
pungent Pungency ( ) is the taste of food commonly referred to as spiciness, hotness or heat, found in foods such as chili peppers. Highly pungent tastes may be experienced as unpleasant. The term piquancy ( ) is sometimes applied to foods with a lower ...
,
astringent An astringent (sometimes called adstringent) is a chemical that shrinks or constricts body tissues. The word derives from the Latin '' adstringere'', which means "to bind fast". Astringency, the dry, puckering or numbing mouthfeel caused by t ...
, sour,
salt In common usage, salt is a mineral composed primarily of sodium chloride (NaCl). When used in food, especially in granulated form, it is more formally called table salt. In the form of a natural crystalline mineral, salt is also known as r ...
, and bitter). The Bhagavati is also the first text to mention the choose function. In the second century BC,
Pingala Acharya Pingala (; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (), also called the ''Pingala-sutras'' (), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the ...
included an enumeration problem in the Chanda Sutra (also Chandahsutra) which asked how many ways a six-syllable
meter The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
could be made from short and long notes. Pingala found the number of meters that had n long notes and k short notes; this is equivalent to finding the
binomial coefficients 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 te ...
. The ideas of the Bhagavati were generalized by the Indian mathematician
Mahavira Mahavira (Devanagari: महावीर, ), also known as Vardhamana (Devanagari: वर्धमान, ), was the 24th ''Tirthankara'' (Supreme Preacher and Ford Maker) of Jainism. Although the dates and most historical details of his lif ...
in 850 AD, and Pingala's work on prosody was expanded by
Bhāskara II Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
and Hemacandra in 1100 AD. Bhaskara was the first known person to find the generalised
choice function Let ''X'' be a set of sets none of which are empty. Then a choice function (selector, selection) on ''X'' is a mathematical function ''f'' that is defined on ''X'' such that ''f'' is a mapping that assigns each element of ''X'' to one of its ele ...
, although
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
may have known earlier. Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the
Fibonacci numbers In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...
. The ancient Chinese book of divination
I Ching The ''I Ching'' or ''Yijing'' ( ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. The ''I Ching'' was originally a divination manual in ...
describes a
hexagram , can be seen as a compound polygon, compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram (Greek language, Greek) or sexagram (Latin l ...
as a permutation with repetitions of six lines where each line can be one of two states: solid or dashed. In describing hexagrams in this fashion they determine that there are 2^6=64 possible hexagrams. A Chinese
monk A monk (; from , ''monachos'', "single, solitary" via Latin ) is a man who is a member of a religious order and lives in a monastery. A monk usually lives his life in prayer and contemplation. The concept is ancient and can be seen in many reli ...
also may have counted the number of configurations to a game similar to Go around 700 AD. Although China had relatively few advancements in enumerative combinatorics, around 100 AD they solved the
Lo Shu Square The Luoshu (pinyin), Lo Shu (Wade-Giles), or Nine Halls Diagram is an Ancient China, ancient Chinese diagram and named for the Luo River (Henan), Luo River near Luoyang, Henan. The Luoshu appears in Chinese mythology, myths concerning the Chinese ...
which is the
combinatorial design Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These co ...
problem of the normal
magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
of order three. Magic squares remained an interest of China, and they began to generalize their original 3\times3 square between 900 and 1300 AD. China corresponded with the
Middle East The Middle East (term originally coined in English language) is a geopolitical region encompassing the Arabian Peninsula, the Levant, Turkey, Egypt, Iran, and Iraq. The term came into widespread usage by the United Kingdom and western Eur ...
about this problem in the 13th century. The Middle East also learned about
binomial coefficients 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 te ...
from Indian work and found the connection to
polynomial expansion In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that ...
. The work of
Hindus Hindus (; ; also known as Sanātanīs) are people who religiously adhere to Hinduism, also known by its endonym Sanātana Dharma. Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pp. 35–37 Historically, the term has also be ...
influenced
Arabs Arabs (,  , ; , , ) are an ethnic group mainly inhabiting the Arab world in West Asia and North Africa. A significant Arab diaspora is present in various parts of the world. Arabs have been in the Fertile Crescent for thousands of yea ...
as seen in the work of al-Khalil ibn Ahmad who considered the possible arrangements of letters to form
syllables A syllable is a basic unit of organization within a sequence of speech sounds, such as within a word, typically defined by linguists as a ''nucleus'' (most often a vowel) with optional sounds before or after that nucleus (''margins'', which are ...
. His calculations show an understanding of
permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...
and
combinations In mathematics, a combination is a selection of items from a set that has distinct members, 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 t ...
. In a passage from the work of Arab mathematician Umar al-Khayyami that dates to around 1100, it is corroborated that the Hindus had knowledge of binomial coefficients, but also that their methods reached the middle east. Abū Bakr ibn Muḥammad ibn al Ḥusayn Al-Karaji (c. 953–1029) wrote on the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
and
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
. In a now lost work known only from subsequent quotation by al-Samaw'al,
Al-Karaji (; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on ...
introduced the idea of argument by
mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold. This is done by first proving a ...
. The
philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
and
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
Rabbi A rabbi (; ) is a spiritual leader or religious teacher in Judaism. One becomes a rabbi by being ordained by another rabbi—known as ''semikha''—following a course of study of Jewish history and texts such as the Talmud. The basic form of t ...
Abraham ibn Ezra Abraham ben Meir Ibn Ezra (, often abbreviated as ; ''Ibrāhim al-Mājid ibn Ezra''; also known as Abenezra or simply ibn Ezra, 1089 / 1092 – 27 January 1164 / 23 January 1167)''Jewish Encyclopedia''online; '' Chambers Biographical Dictionar ...
(c. 1140) counted the permutations with repetitions in vocalization of Divine Name. He also 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 The Talmud (; ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of Haskalah#Effects, modernity, in nearly all Jewish communities, the Talmud was the cen ...
and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Levi ben Gerson (better known as Gersonides), in 1321. The arithmetical triangle—a graphical diagram showing relationships among the
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— was presented by mathematicians in
treatises A treatise is a Formality, formal and systematic written discourse on some subject concerned with investigating or exposing the main principles of the subject and its conclusions."mwod:treatise, Treatise." Merriam-Webster Online Dictionary. Acc ...
dating as far back as the 10th century, and would eventually become known as
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
. Later, in 17th century England,
campanology Campanology (/kæmpəˈnɒlədʒi/) is both the scientific and artistic study of bells, encompassing their design, tuning, and the methods by which they are rung. It delves into the technology behind bell casting and tuning, as well as the rich ...
provided examples of what is now known as
Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
s in certain
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...
s on
permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...
.


Combinatorics in the West

Combinatorics came to
Europe Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. It is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, the Mediterranean Sea to the south, and Asia to the east ...
in the 13th century through mathematicians Leonardo Fibonacci and Jordanus de Nemore. Fibonacci's
Liber Abaci The or (Latin for "The Book of Calculation") was a 1202 Latin work on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. It is primarily famous for introducing both base-10 positional notation and the symbols known as Arabic n ...
introduced many of the
Arabian The Arabian Peninsula (, , or , , ) or Arabia, is a peninsula in West Asia, situated north-east of Africa on the Arabian plate. At , comparable in size to India, the Arabian Peninsula is the largest peninsula in the world. Geographically, the ...
and Indian ideas to Europe, including that of the
Fibonacci numbers In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...
. Jordanus was the first person to arrange the binomial coefficients in a
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, as he did in
proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
70 of ''De Arithmetica''. This was also done in the
Middle East The Middle East (term originally coined in English language) is a geopolitical region encompassing the Arabian Peninsula, the Levant, Turkey, Egypt, Iran, and Iraq. The term came into widespread usage by the United Kingdom and western Eur ...
in 1265, and
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
around 1300. Today, this triangle is known as
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
. Pascal's contribution to the triangle that bears his name comes from his work on formal proofs about it, and the connections he made between Pascal's triangle and
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
. From a letter
Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...
sent to
Daniel Bernoulli Daniel Bernoulli ( ; ; – 27 March 1782) was a Swiss people, Swiss-France, French mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applicati ...
we learn that Leibniz was formally studying the mathematical theory of partitions in the 17th century, although no formal work was published. Together with Leibniz, Pascal published De Arte Combinatoria in 1666 which was reprinted later. Pascal and Leibniz are considered the founders of modern combinatorics. Both Pascal and Leibniz understood that the binomial expansion was equivalent to the
choice function Let ''X'' be a set of sets none of which are empty. Then a choice function (selector, selection) on ''X'' is a mathematical function ''f'' that is defined on ''X'' such that ''f'' is a mapping that assigns each element of ''X'' to one of its ele ...
. The notion that
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and combinatorics corresponded was expanded by De Moivre, who found the expansion of a multinomial. De Moivre also found the formula for derangements using the principle of principle of inclusion–exclusion, a method different from Nikolaus Bernoulli, who had found it previously. De Moivre also managed to approximate the
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 and
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
, and found a closed form for the
Fibonacci numbers In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...
by inventing
generating functions In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
. In the 18th century,
Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
worked on problems of combinatorics, and several problems of
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
which are linked to combinatorics. Problems Euler worked on include the Knights tour, Graeco-Latin square, Eulerian numbers, and others. To solve the
Seven Bridges of Königsberg The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in Prussia ...
problem he invented
graph theory In mathematics and computer science, 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 of ''Vertex (graph ...
, which also led to the formation of
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
. Finally, he broke ground with partitions by the use of
generating functions In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
.


Contemporary combinatorics

In the 19th century, the subject of
partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
s and
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
originated in the work of Dedekind, Peirce, and Schröder. However, it was
Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician Ge ...
's seminal work in his book ''Lattice Theory'' published in 1967, and the work of
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
that truly established the subjects. In the 1930s,
Hall In architecture, a hall is a relatively large space enclosed by a roof and walls. In the Iron Age and the Early Middle Ages in northern Europe, a mead hall was where a lord and his retainers ate and also slept. Later in the Middle Ages, the gre ...
(1936) and Weisner (1935) independently stated the general
Möbius inversion formula In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large genera ...
. In 1964, Gian-Carlo Rota's ''On the Foundations of Combinatorial Theory I. Theory of Möbius Functions '' introduced
poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
and lattice theory as theories in Combinatorics. Richard P. Stanley has had a big impact in contemporary combinatorics for his work in
matroid theory In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms ...
, for introducing Zeta polynomials, for explicitly defining Eulerian posets, developing the theory of binomial posets along with Rota and Peter Doubilet, and more.
Paul Erdős Paul Erdős ( ; 26March 191320September 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 discrete mathematics, g ...
made seminal contributions to combinatorics throughout the century, winning the
Wolf prize The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for "achievements in the interest of mankind and friendly relations among people ... irrespective of natio ...
in-part for these contributions.


Notes


References

* N.L. Biggs, The roots of combinatorics, ''Historia Mathematica'' 6 (1979), 109–136. * Katz, Victor J. (1998). ''A History of Mathematics: An Introduction'', 2nd Edition. Addison-Wesley Education Publishers. . * O'Connor, John J. and Robertson, Edmund F. (1999–2004). '' MacTutor History of Mathematics archive''.
St Andrews University The University of St Andrews (, ; abbreviated as St And in post-nominals) is a public university in St Andrews, Scotland. It is the oldest of the four ancient universities of Scotland and, following the universities of Oxford and Cambridge, t ...
. * Rashed, R. (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. London. * Wilson, R. and Watkins, J. (2013). ''Combinatorics: Ancient & Modern''. Oxford. * Stanley, Richard (2012). ''Enumerative combinatorics (2nd ed. ed.)'', 2nd Edition. Cambridge University Press. . {{DEFAULTSORT:History Of Combinatorics
Combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
+