View source for Combinatorics

{{Short description|Branch of discrete mathematics}}
{{Distinguish|Combinatoriality}}
{{Redirect | Combinatorial | combinatorial logic in computer science | Combinatorial logic }}
{{More footnotes needed|date=July 2022}}
{{Math topics TOC|expanded=Pure mathematics}}
'''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 set|finite]] [[Mathematical structure|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]], notably in [[algebra]], [[probability theory]], [[topology]], and [[geometry]],<ref>Björner and Stanley, p. 2</ref> 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.<ref>{{cite book|last1=Lovász|first1=László|title=Combinatorial Problems and Exercises|date=1979|publisher=North-Holland|quote=In my opinion, combinatorics is now growing out of this early stage.|url=https://books.google.com/books?id=ueq1CwAAQBAJ&pg=PP1|isbn=978-0821842621|access-date=2021-03-23|archive-date=2021-04-16|archive-url=https://web.archive.org/web/20210416100842/https://books.google.com/books?id=ueq1CwAAQBAJ&pg=PP1|url-status=live}}</ref> 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]].

== Definition ==
The full scope of combinatorics is not universally agreed upon.<ref>{{cite web |last=Pak |first=Igor |title=What is Combinatorics? |url=https://www.math.ucla.edu/~pak/hidden/papers/Quotes/Combinatorics-quotes.htm |access-date=1 November 2017 |archive-date=17 October 2017 |archive-url=https://web.archive.org/web/20171017075155/http://www.math.ucla.edu/~pak/hidden/papers/Quotes/Combinatorics-quotes.htm |url-status=live }}</ref> According to [[H. J. Ryser]], a definition of the subject is difficult because it crosses so many mathematical subdivisions.<ref>{{harvnb|Ryser|1963|loc=p. 2}}</ref> 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''.

According to [[Leon Mirsky]], "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."<ref>{{citation |last=Mirsky |first=Leon |title=Book Review |url=https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979-14606-8/S0273-0979-1979-14606-8.pdf |journal=Bulletin of the American Mathematical Society |volume=1 |pages=380–388 |year=1979 |series=New Series |doi=10.1090/S0273-0979-1979-14606-8 |doi-access=free |access-date=2021-02-04 |archive-date=2021-02-26 |archive-url=https://web.archive.org/web/20210226080424/https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979-14606-8/S0273-0979-1979-14606-8.pdf |url-status=live }}</ref> 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.<ref>{{cite book |last1=Rota |first1=Gian Carlo |url=https://link.springer.com/book/10.1007/978-0-8176-4775-9 |title=Discrete Thoughts |date=1969 |publisher=Birkhaüser |isbn=978-0-8176-4775-9 |page=50 |doi=10.1007/978-0-8176-4775-9 |quote=... combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent ... . The typical ... case of this is algebraic topology (formerly known as combinatorial topology)}}</ref> Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, [[Countable set|countable]]) but [[Discrete mathematics|discrete]] setting.

== History ==
[[Image:Plain-bob-minor 2.png|150px|right|thumb|An example of [[change ringing]] (with six bells), one of the earliest nontrivial results in [[graph theory]].]]

{{Main|History of combinatorics}}

Basic combinatorial concepts and enumerative results appeared throughout the [[Ancient history|ancient world]]. The earliest recorded use of combinatorial techniques comes from problem 79 of the [[Rhind papyrus]], which dates to the 16th century BC.  The problem concerns a certain [[geometric series]], and has similarities to Fibonacci's problem of counting the number of [[composition (combinatorics)|compositions]] of 1s and 2s that [[Summation|sum]] to a given total.<ref name="Biggs">{{cite book
| last1 = Biggs
| first1 = Norman
| first2 = Keith 
| last2= Lloyd 
| first3 = Robin
| last3 = Wilson
 | editor = Ronald Grahm, Martin Grötschel, László Lovász
 | title = Handbook of Combinatorics
 | year = 1995
 | url = https://books.google.com/books?id=kfiv_-l2KyQC
 | format = Google book
 | accessdate = 2008-03-08
 | publisher = MIT Press
 | isbn = 0262571722
 | pages = 2163–2188
 | chapter = 44
 }}</ref> [[Timeline of Indian history|Indian]] [[physician]] [[Sushruta]] asserts in [[Sushruta Samhita]] that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 2<sup>6</sup>&nbsp;−&nbsp;1 possibilities.  [[Ancient Greece|Greek]] [[historian]] [[Plutarch]] discusses an argument between [[Chrysippus]] (3rd century BCE) and [[Hipparchus]] (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to [[Schröder–Hipparchus number]]s.<ref>{{Cite journal|last=Acerbi|first=F.|title=On the shoulders of Hipparchus|url=https://link.springer.com/article/10.1007/s00407-003-0067-0|journal=Archive for History of Exact Sciences|year=2003|volume=57|issue=6|pages=465–502|doi=10.1007/s00407-003-0067-0|s2cid=122758966|access-date=2021-03-12|archive-date=2022-01-23|archive-url=https://web.archive.org/web/20220123111759/https://link.springer.com/article/10.1007%2Fs00407-003-0067-0|url-status=live|url-access=subscription}}</ref><ref>[[Richard P. Stanley|Stanley, Richard P.]]; "Hipparchus, Plutarch, Schröder, and Hough", ''American Mathematical Monthly'' '''104''' (1997), no. 4, 344–350.</ref><ref>{{cite journal |doi=10.1080/00029890.1998.12004906 |title=On the Second Number of Plutarch |year=1998 |last1=Habsieger |first1=Laurent |last2=Kazarian |first2=Maxim |last3=Lando |first3=Sergei |journal=The American Mathematical Monthly |volume=105 |issue=5 |pages=446 }}</ref> Earlier, in the ''[[Ostomachion]]'', [[Archimedes]] (3rd century BCE) may have considered the number of configurations of a [[tiling puzzle]],<ref>{{Cite journal|last1=Netz|first1=R.|last2=Acerbi|first2=F.|last3=Wilson|first3=N.|title=Towards a reconstruction of Archimedes' Stomachion|url=https://www.sciamvs.org/2004.html|journal=Sciamvs|volume=5|pages=67–99|access-date=2021-03-12|archive-date=2021-04-16|archive-url=https://web.archive.org/web/20210416095205/https://www.sciamvs.org/2004.html|url-status=live}}</ref> while combinatorial interests possibly were present in lost works by [[Apollonius of Perga|Apollonius]].<ref>{{Cite journal|last=Hogendijk|first=Jan P.|date=1986|title=Arabic Traces of Lost Works of Apollonius|url=https://www.jstor.org/stable/41133783|journal=Archive for History of Exact Sciences|volume=35|issue=3|pages=187–253|doi=10.1007/BF00357307|jstor=41133783|s2cid=121613986|issn=0003-9519|access-date=2021-03-26|archive-date=2021-04-18|archive-url=https://web.archive.org/web/20210418221029/https://www.jstor.org/stable/41133783|url-status=live}}</ref><ref>{{Cite journal|last=Huxley|first=G.|date=1967|title=Okytokion|url=https://grbs.library.duke.edu/article/view/11131/4205|journal=Greek, Roman, and Byzantine Studies|volume=8|issue=3|pages=203|access-date=2021-03-26|archive-date=2021-04-16|archive-url=https://web.archive.org/web/20210416100956/https://grbs.library.duke.edu/article/view/11131/4205|url-status=live}}</ref>

In the [[Middle Ages]], combinatorics continued to be studied, largely outside of the [[Culture of Europe|European civilization]]. The [[India]]n mathematician [[Mahāvīra (mathematician)|Mahāvīra]] ({{circa|850}}) provided formulae for the number of [[permutation]]s and [[combination]]s,<ref>{{MacTutor | id=Mahavira}}</ref><ref>{{cite book |last=Puttaswamy |first=Tumkur K. |contribution=The Mathematical Accomplishments of Ancient Indian Mathematicians |editor-last=Selin |editor-first=Helaine |title=Mathematics Across Cultures: The History of Non-Western Mathematics |publisher=Kluwer Academic Publishers |location=Netherlands |url=https://books.google.com/books?id=2hTyfurOH8AC |year=2000 |isbn=978-1-4020-0260-1 |page=417 <!-- pages 409–422 --> |access-date=2015-11-15 |archive-date=2021-04-16 |archive-url=https://web.archive.org/web/20210416100852/https://books.google.com/books?id=2hTyfurOH8AC |url-status=live }}</ref> and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.<ref>{{cite journal | last1 = Biggs | first1 = Norman L. | year = 1979 | title = The Roots of Combinatorics | journal = Historia Mathematica | volume = 6 | issue = 2| pages = 109–136 | doi=10.1016/0315-0860(79)90074-0| doi-access = free }}</ref>  The [[philosopher]] and [[astronomer]] Rabbi [[Abraham ibn Ezra]] ({{circa|1140}}) established the symmetry of [[binomial coefficient]]s, while a closed formula was obtained later by the [[talmudist]] and [[mathematician]] [[Levi ben Gerson]] (better known as Gersonides), in 1321.<ref>{{citation|title=Probability Theory: A Historical Sketch|first=L.E.|last=Maistrov|publisher=Academic Press|year=1974|isbn=978-1-4832-1863-2|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PA35|page=35|access-date=2015-01-25|archive-date=2021-04-16|archive-url=https://web.archive.org/web/20210416091311/https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PA35|url-status=live}}. (Translation from 1967 Russian ed.)</ref>
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 cycle]]s in certain [[Cayley graph]]s on permutations.<ref>{{cite journal |doi=10.1080/00029890.1987.12000711 |title=Ringing the Cosets |year=1987 |last1=White |first1=Arthur T. |journal=The American Mathematical Monthly |volume=94 |issue=8 |pages=721–746 }}</ref><ref>{{cite journal |doi=10.1080/00029890.1996.12004816 |title=Fabian Stedman: The First Group Theorist? |year=1996 |last1=White |first1=Arthur T. |journal=The American Mathematical Monthly |volume=103 |issue=9 |pages=771–778 }}</ref>

During the [[Renaissance]], together with the rest of mathematics and the [[science]]s, combinatorics enjoyed a rebirth. Works of [[Blaise Pascal|Pascal]], [[Isaac Newton|Newton]], [[Jacob Bernoulli]] and [[Leonhard Euler|Euler]] became foundational in the emerging field. In modern times, the works of [[James Joseph Sylvester|J.J. Sylvester]] (late 19th century) and [[Percy Alexander MacMahon|Percy MacMahon]] (early 20th century) helped lay the foundation for [[Enumerative combinatorics|enumerative]] and [[algebraic combinatorics]]. [[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.<ref>See [http://www.math.iit.edu/~kaul/Journals.html#CGT Journals in Combinatorics and Graph Theory] {{Webarchive|url=https://web.archive.org/web/20210217150357/http://www.math.iit.edu/~kaul/Journals.html#CGT |date=2021-02-17 }}</ref> 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===
[[Image:Catalan 4 leaves binary tree example.svg|320px|right|thumb|Five [[binary tree]]s on three [[Vertex (graph theory)|vertices]], an example of [[Catalan number]]s.]]

{{Main|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]], 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 [[Partition of a set|partitions]].

===Analytic combinatorics===
{{Main|Analytic combinatorics}}
[[Analytic combinatorics]] concerns the enumeration of combinatorial structures using tools from [[complex analysis]] 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 analysis|asymptotic formulae]].

=== Partition theory ===
[[Image:Partition3D.svg|150px|right|thumb|A [[plane partition]].]]

{{Main|Integer partition|l1=Partition theory}}

Partition theory studies various enumeration and asymptotic problems related to [[integer partition]]s, 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 proof|bijective approach]] and various tools in analysis and [[analytic number theory]] and has connections with [[statistical mechanics]]. Partitions can be graphically visualized with [[Young diagram]]s or [[Ferrers diagram]]s. They occur in a number of branches of [[mathematics]] and [[physics]], including the study of [[symmetric polynomial]]s and of the [[symmetric group]] and in [[Group representation|group representation theory]] in general.

===Graph theory===
[[Image:Petersen1 tiny.svg|thumb|150px|[[Petersen graph]].]]
{{Main|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''<sub>''G''</sub>(''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.<ref>Sanders, Daniel P.; [http://www.math.gatech.edu/~sanders/graphtheory/writings/2-digit.html ''2-Digit MSC Comparison''] {{webarchive|url=https://web.archive.org/web/20081231163112/http://www.math.gatech.edu/~sanders/graphtheory/writings/2-digit.html |date=2008-12-31 }}</ref>  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===
{{Main|Combinatorial design}}

Design theory is a study of [[combinatorial design]]s, which are collections of subsets with certain [[Set intersection|intersection]] properties. [[Block design]]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 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|tournament scheduling]], [[Lottery|lotteries]], [[mathematical chemistry]], [[mathematical biology]], [[Algorithm design|algorithm design and analysis]], [[Computer network|networking]], [[group testing]] and [[cryptography]].<ref>{{harvnb|Stinson|2003|loc=pg.1}}</ref>

===Finite geometry===
{{Main|Finite geometry}}

Finite geometry is the study of [[Geometry|geometric systems]] having only a finite number of points. Structures analogous to those found in continuous geometries ([[Euclidean plane]], [[real projective space]], etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for [[Combinatorial design|design theory]]. It should not be confused with discrete geometry ([[combinatorial geometry]]).

=== Order theory ===
[[Image:Hasse diagram of powerset of 3.svg|thumb|right|150px|[[Hasse diagram]] of the [[Power set|powerset]] of {x,y,z} ordered by [[Inclusion map|inclusion]].]]

{{Main|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 [[abstract algebra|algebra]], geometry, number theory and throughout combinatorics and graph theory.  Notable classes and examples of partial orders include [[Lattice (order)|lattices]] and [[Boolean algebras]].

===Matroid theory===
{{Main|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 independence|linear dependence]] 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===
{{Main|Extremal combinatorics}}Extremal combinatorics studies how large or how small a collection of finite objects ([[number]]s, [[Graph (discrete mathematics)|graphs]], [[Vector space|vectors]], [[Set (mathematics)|sets]], etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns [[Class (set theory)|classes]] of [[set system]]s; this is called extremal set theory. For instance, in an ''n''-element set, what is the largest number of ''k''-element [[subset]]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 family#Sperner's theorem|Sperner's theorem]], 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<sub>n,n</sub>''. Often it is too hard even to find the extremal answer ''f''(''n'') exactly and one can only give an [[asymptotic analysis|asymptotic estimate]].

[[Ramsey theory]] is another part of extremal combinatorics.  It states that any [[sufficiently large]] configuration will contain some sort of order. It is an advanced generalization of the [[pigeonhole principle]].

===Probabilistic combinatorics===
[[Image:Self avoiding walk.svg|thumb|right|150px|[[Self-avoiding walk]] in a [[Lattice graph|square grid graph]].]]

{{Main|Probabilistic method}}

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 [[Markov chain mixing time|mixing time]].{{clarify|date=November 2022}}

Often associated with [[Paul Erdős]], 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. The area recently grew to become an independent field of combinatorics.

===Algebraic combinatorics===
[[Image:Young diagram for 541 partition.svg|thumb|right|150px|[[Young diagram]] of the [[integer partition]] (5, 4, 1).]]
{{Main|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 [[abstract algebra|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 [[Enumerative combinatorics|enumerative]] in nature or involve [[matroid]]s, [[polytope]]s, [[partially ordered set]]s, or [[Finite geometry|finite geometries]]. On the algebraic side, besides group and representation theory, [[lattice theory]] and [[commutative algebra]] are common.

===Combinatorics on words===
[[Image:Morse-Thue sequence.gif|thumb|right|210px|Construction of a [[Thue–Morse sequence|Thue–Morse infinite word]].]]

{{Main|Combinatorics on words}}

Combinatorics on words deals with [[formal language]]s.  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]] of classes of [[formal grammar]]s is perhaps the best-known result in the field.

===Geometric combinatorics===
[[Image:Icosahedron.svg|150px|thumb|right|An [[icosahedron]].]]

{{Main|Geometric combinatorics}}

Geometric combinatorics is related to [[Convex geometry|convex]] and [[discrete geometry]]. 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 polytope]]s. [[Combinatorial geometry]] is a historical name for discrete geometry.

It includes a number of subareas such as [[polyhedral combinatorics]] (the study of [[Face (geometry)|faces]] of [[Convex polyhedron|convex polyhedra]]), [[convex geometry]] (the study of [[convex set]]s, in particular combinatorics of their intersections), and [[discrete geometry]], which in turn has many applications to [[computational geometry]]. The study of [[regular polytope]]s, [[Archimedean solid]]s, and [[kissing number]]s is also a part of geometric combinatorics. Special polytopes are also considered, such as the [[permutohedron]], [[associahedron]] and [[Birkhoff polytope]].

===Topological combinatorics===
[[Image:Collier-de-perles-rouge-vert.svg|150px|thumb|right|[[Necklace splitting problem|Splitting a necklace]] with two cuts.]]
{{Main|Topological combinatorics}}

Combinatorial analogs of concepts and methods in [[topology]] are used to study [[graph coloring]], [[fair division]], [[partition of a set|partitions]], [[partially ordered set]]s, [[decision tree]]s, [[necklace problem]]s and [[discrete Morse theory]].  It should not be confused with [[combinatorial topology]] which is an older name for [[algebraic topology]].

===Arithmetic combinatorics===
{{Main|Arithmetic combinatorics}}

Arithmetic combinatorics arose out of the interplay between [[number theory]], 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 system]]s.

===Infinitary combinatorics===
{{Main|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 graph]]s and [[Tree (set theory)|trees]], extensions of [[Ramsey's theorem]], and [[Martin's axiom]]. Recent developments concern combinatorics of the [[Continuum (set theory)|continuum]]<ref>[[Andreas Blass]], ''Combinatorial Cardinal Characteristics of the Continuum'', Chapter 6 in Handbook of Set Theory, edited by [[Matthew Foreman]] and [[Akihiro Kanamori]], Springer, 2010</ref> and combinatorics on successors of singular cardinals.<ref>{{Citation |last=Eisworth |first=Todd |title=Successors of Singular Cardinals |date=2010 |url=http://link.springer.com/10.1007/978-1-4020-5764-9_16 |work=Handbook of Set Theory |pages=1229–1350 |editor-last=Foreman |editor-first=Matthew |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-1-4020-5764-9_16 |isbn=978-1-4020-4843-2 |access-date=2022-08-27 |editor2-last=Kanamori |editor2-first=Akihiro|url-access=subscription }}</ref>

[[Gian-Carlo Rota]] used the name ''continuous combinatorics''<ref>{{Cite web |url=http://faculty.uml.edu/dklain/cpc.pdf |title=''Continuous and profinite combinatorics'' |access-date=2009-01-03 |archive-date=2009-02-26 |archive-url=https://web.archive.org/web/20090226040144/http://faculty.uml.edu/dklain/cpc.pdf |url-status=live }}</ref> to describe [[geometric probability]], since there are many analogies between ''counting'' and ''measure''.

== Related fields ==
[[Image:Kissing-3d.png|150px|right|thumb|[[kissing number|Kissing spheres]] are connected to both [[coding theory]] and [[discrete geometry]].]]

=== 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 code]]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]].

=== Discrete and computational geometry ===
[[Discrete geometry]] (also called combinatorial geometry) also began as a part of combinatorics, with early results on [[convex polytope]]s and [[kissing number]]s.  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]], 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 polynomial|chromatic]] and [[Tutte polynomial]]s on the other hand.

==See also==
{{Portal|Mathematics}}
{{cols}}
* [[Combinatorial biology]]
* [[Combinatorial chemistry]]
* [[Combinatorial data analysis]]
* [[Combinatorial game theory]]
* [[Combinatorial group theory]]
* [[Discrete mathematics]]
* [[List of combinatorics topics]]
* [[Phylogenetics]]
* [[Polynomial method in combinatorics]]
{{Div col end}}

== Notes ==
{{Reflist}}

==References==
* Björner, Anders; and Stanley, Richard P.; (2010); [http://www-math.mit.edu/~rstan/papers/comb.pdf ''A Combinatorial Miscellany'']
* Bóna, Miklós; (2011); [http://www.worldscientific.com/worldscibooks/10.1142/8027 ''A Walk Through Combinatorics (3rd ed.)'']. {{ISBN|978-981-4335-23-2|978-981-4460-00-2}}
* 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. {{ISBN|0-262-07169-X}}
* Lindner, Charles C.; and Rodger, Christopher A.; eds. (1997); ''Design Theory'', CRC-Press. {{ISBN|0-8493-3986-3}}.
* {{citation|first=John|last=Riordan|author-link=John Riordan (mathematician)|orig-year= 1958|year=2002|title=An Introduction to Combinatorial Analysis|publisher=Dover|isbn=978-0-486-42536-8}}
* {{citation|first=Herbert John|last=Ryser|title=Combinatorial Mathematics|series=The Carus Mathematical Monographs(#14)|year=1963|publisher=The Mathematical Association of America}}
* [[Richard P. Stanley|Stanley, Richard P.]] (1997, 1999); [http://www-math.mit.edu/~rstan/ec/ ''Enumerative Combinatorics'', Volumes 1 and 2], [[Cambridge University Press]]. {{ISBN|0-521-55309-1|0-521-56069-1}}
* {{citation |last=Stinson |first=Douglas R. |title=Combinatorial Designs: Constructions and Analysis |year=2003 |publisher=Springer |location=New York |isbn=0-387-95487-2}}
* van Lint, Jacobus H.; and Wilson, Richard M.; (2001); ''A Course in Combinatorics'', 2nd ed., Cambridge University Press. {{ISBN|0-521-80340-3}}

== External links ==
{{Sister project links| wikt=combinatorics | commons=Category:Combinatorics | b=Probability/Combinatorics | n=no | q=Combinatorics | s=no | v=no | voy=no | species=no | d=no}}
* {{springer|title=Combinatorial analysis|id=p/c023250}}
* [https://www.britannica.com/science/combinatorics Combinatorial Analysis] – an article in [[Encyclopædia Britannica Eleventh Edition]]
* [http://mathworld.wolfram.com/Combinatorics.html Combinatorics], a [[MathWorld]] article with many references.
* [http://www.mathpages.com/home/icombina.htm Combinatorics], from a ''MathPages.com'' portal.
* [http://www.combinatorics.net/Resources/hyper/Hyperbook.aspx The Hyperbook of Combinatorics], a collection of math articles links.
* [http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf The Two Cultures of Mathematics] by W.T. Gowers, article on problem solving vs theory building.
* [http://www.math.illinois.edu/~dwest/openp/gloss.html "Glossary of Terms in Combinatorics"] {{Webarchive|url=https://web.archive.org/web/20170817044031/http://www.math.illinois.edu/~dwest/openp/gloss.html |date=2017-08-17 }}
* [https://www.mat.univie.ac.at/~slc/divers/software.html List of Combinatorics Software and Databases]
{{Factorial and binomial topics}}
{{Areas of mathematics}}
{{Computer science}}
{{Authority control}}

[[Category:Combinatorics| ]]