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Branches of Mathematic

Combinatorics

Also Known As Combinatorial Mathematics
Combinatorics and Graph Theory

The branch of mathematics concerned with counting, arrangement and the existence of discrete, finite structures: how many ways an alphabet's letters can be permuted, a set can be partitioned, a graph can be colored, or a combinatorial design can be built. Its methods range from elementary counting arguments to deep tools such as generating functions and the probabilistic method. Paul Erdos, working across enumerative, extremal and probabilistic combinatorics for most of the twentieth century, did more than any other single mathematician to establish combinatorics as a field of its own rather than a bag of isolated puzzles, publishing with hundreds of collaborators and helping found extremal graph theory and Ramsey theory. Frank Ramsey's own 1930 paper, on a question in formal logic, contains as a lemma the result now called Ramsey's theorem, one of combinatorics' most cited results, showing that sufficiently large structures cannot avoid all order: complete disorder is impossible.

Facts
Central Question
How many ways can a finite structure be arranged, counted or shown to exist at all, and can that number, or even just its existence, be determined without listing every case? 2
Key Debate
Whether combinatorics is a unified branch of mathematics with its own deep theory, or a loose collection of clever, ad hoc arguments unified mainly by subject matter, finite discrete structures, rather than by shared method, a charge sometimes leveled at the field by mathematicians from more axiomatically unified branches. Combinatorialists reject the charge by pointing to genuinely deep, general tools, the probabilistic method, generating functions, and algebraic and topological methods, now used throughout the field. 2
Classification
Pure or Applied
Both / Interdisciplinary 1
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The Man Who Had Hundreds of Coauthors

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Paul Erdos owned almost nothing and lived almost nowhere. He carried a suitcase between the homes of mathematicians around the world, arrived unannounced, said Let n be an integer, and started working. What he left behind was not a single great theorem but a way of doing an entire field: combinatorics, the mathematics of counting, arranging and finding order inside finite structures, had existed before him as a scattering of clever individual results. Erdos, working with whichever mathematician was in the room, turned it into a single connected conversation. He wrote more than 1,500 papers with more than 500 different coauthors, so many that mathematicians now compute their own Erdos number, the length of the shortest coauthorship chain back to him, half joke and half genuine record of how far his collaborative habit reached. Extremal graph theory, which asks how large or small a graph can be while still avoiding some structure, and the modern form of Ramsey theory, which asks how much order is forced on a large enough structure no matter how it is built, both took their working shape substantially through Erdos and the mathematicians he worked beside. He also pioneered the probabilistic method: to prove that a structure with some property must exist, show that a randomly built structure has that property with positive probability, so at least one example does, without ever constructing it by hand. It is now one of combinatorics' standard tools, used across the field for problems that resist direct construction entirely. Erdos never held a permanent academic position for long and gave away most of the money from the prizes he won, offering small cash rewards of his own for problems he could not solve himself, some of which still stand unclaimed.

Complete Disorder Is Impossible

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

In 1928, a young English mathematician and philosopher named Frank Ramsey was not trying to found a branch of combinatorics. He was trying to settle a question in formal logic, the decision problem for a fragment of first-order logic, and along the way he proved a lemma so useful on its own that it eventually outgrew the paper that contained it. The result now called Ramsey's theorem says, in its simplest form, that if the connections between enough objects are each colored one of two colors, some large fully one-colored group of connections is guaranteed to exist, no matter how the coloring is chosen. Color every pair of people at a large enough party as mutual acquaintances or mutual strangers, and Ramsey's theorem guarantees a group of a certain size who all know each other, or a group of a certain size who are all strangers, however the party's friendships happen to fall. The theorem generalizes far beyond parties and pairs, to larger groups, more colors and more abstract objects than people, but the party version is the one that made the result famous outside mathematics. Ramsey died in 1930 at twenty-six, having never seen his lemma become a field. Ramsey theory, the branch of combinatorics built on generalizing it, asks how large a structure has to be before some specified kind of order becomes unavoidable inside it, and the honest, almost paradoxical answer his theorem gives is that past a certain size, complete disorder is not a possibility at all. The exact size where order becomes unavoidable, the Ramsey number for a given case, is itself often extremely hard to compute; some small Ramsey numbers are still not known exactly, only bounded, decades after the theorem that guarantees their existence.

Combinatorics
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Connections

Associated With

Source Paul Erdos (1913-1996), Biography (MacTutor History of Mathematics)

Includes

Source Wolfram MathWorld: Abel's Binomial Theorem
Source Alfréd Rényi, Biography (MacTutor History of Mathematics)
Source Aztec diamond (Wikipedia)
Source Balinski's theorem (Wikipedia)
Source Baranyai's theorem (Wikipedia)
Source Behrend's theorem (Wikipedia)
Source Wikipedia: Bell number
Source Wikipedia: Ben Green (mathematician)
Source Bertrand's Ballot Theorem (Wikipedia)
Source Bondy's theorem (Wikipedia)
Source Boolean Pythagorean triples problem, Wikipedia
Source Bregman-Minc inequality (Wikipedia)
Source Bruck-Ryser-Chowla theorem (Wikipedia)
Source Catalan Number (Wikipedia)
Source Catalan Number (Wikipedia)
Source Centered Hexagonal Number (Wikipedia)
Combination, Concepts
Source Combination (Wikipedia)
Source Corners theorem (Wikipedia)
Source Wikipedia: Derangement
Source Wikipedia: Désiré André
Source Dinitz theorem (Wikipedia)
Source Dittert Conjecture (Wikipedia)
Source Dobinski's formula (Wikipedia)
Source Emanuel Sperner (Wikipedia)
Source Endre Szemerédi (Wikipedia)
Source Erdos Conjecture on Arithmetic Progressions (Wikipedia)
Source MacTutor History of Mathematics Archive: Ernst Straus - Biography
Source Eugène Charles Catalan, Biography (MacTutor History of Mathematics)
Source Falling and Rising Factorials (Wikipedia)
Source Fisher's inequality (Wikipedia)
Source Wikipedia: Frank P. Ramsey
Source Freiman's theorem (Wikipedia)
Source Gauss congruence, Wikipedia
Source George Pólya (Wikipedia)
Source George Szekeres, Biography (MacTutor History of Mathematics)
Source Wikipedia: Hao Huang
Source Hirsch Conjecture (Wikipedia)
Source Hook Length Formula (Wikipedia)
Source Issai Schur (Wikipedia)
Source Jacques Ozanam, Biography (MacTutor History of Mathematics)
Source On Coverings by Johanan Schonheim (Pacific Journal of Mathematics, 1964)
Source Kruskal-Katona theorem (Wikipedia)
Source Lambek-Moser theorem (Wikipedia)
Source Lambert W function (Wikipedia)
Source Wikipedia: László Lovász

Showing 1 to 50 of 85 in Includes, page 1 of 2.

Long-Form Articles

Source MacTutor History of Mathematics Archive
Source MacTutor History of Mathematics Archive
Sources
1. Wikipedia: Combinatorics
Wikimedia Foundation
  • Definition section
    Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results.
  • Lead section
    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.
View the Source
2. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statistics
  • https://mathshistory.st-andrews.ac.uk/Extras/Combinatorial_algorithms/
    Combinatorial algorithms are computational procedures which are designed to help solve combinatorial problems.
  • In Category: Branches of Mathematics
  • Includes: Paul Erdos
  • Long-Form Articles: The Man Who Had Hundreds of Coauthors
  • Long-Form Articles: Complete Disorder Is Impossible
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Paul Erdos (1913-1996), Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsAssociated With: Paul Erdos, https://mathshistory.st-andrews.ac.uk/Biographies/Erdos/
Quote, Associated With: Paul Erdos, https://mathshistory.st-andrews.ac.uk/Biographies/Erdos/
founded the field of discrete mathematics
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Union-Closed Sets Conjecture (Wikipedia)
WikipediaIncludes: Union-Closed Sets Conjecture, Lead section
Quote, Includes: Union-Closed Sets Conjecture, Lead section
The union-closed sets conjecture, also known as Frankl's conjecture, is an open problem in combinatorics posed by Péter Frankl in 1979.
View the Source
Lonely Runner Conjecture (Wikipedia)
WikipediaIncludes: Lonely Runner ConjectureView the Source
Pascal's Triangle (Wikipedia)
Wikimedia FoundationIncludes: Pascal's TriangleView the Source
Tutte homotopy theorem (Wikipedia)
Includes: Tutte Homotopy Theorem, Lead
Quote, Includes: Tutte Homotopy Theorem, Lead
generalises the concept of "path" from graphs to matroids
View the Source
Dobinski's formula (Wikipedia)
Includes: Dobinski's Formula, Lead
Quote, Includes: Dobinski's Formula, Lead
In combinatorial mathematics, Dobiński's formula states that the
View the Source
Bregman-Minc inequality (Wikipedia)
Includes: Bregman-Minc Inequality, Lead
Quote, Includes: Bregman-Minc Inequality, Lead
In discrete mathematics, the Bregman-Minc inequality, or Bregman's theorem, allows one to estimate the permanent of a binary matrix
View the Source
MacTutor History of Mathematics Archive: Ernst Straus - Biography
MacTutor History of Mathematics ArchiveIncludes: Ernst G. Straus, Quick Info summary
Quote, Includes: Ernst G. Straus, Quick Info summary
Ernst Straus was a German-born American mathematician who worked in a variety of areas including relativity and combinatorics.
View the Source
On Coverings by Johanan Schonheim (Pacific Journal of Mathematics, 1964)
Includes: Jochanan Schönheim, Introduction
Quote, Includes: Jochanan Schönheim, Introduction
renewed interest has been aroused in the notion of covering and related problems
View the Source
Dittert Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Dittert Conjecture, Lead sentence
Quote, Includes: Dittert Conjecture, Lead sentence
Hajek conjecture, is a mathematical hypothesis in combinatorics concerning the maximum achieved by a particular function ϕ of matr
View the Source
Corners theorem (Wikipedia)
Includes: Corners Theorem, Lead sentenceView the Source
Freiman's theorem (Wikipedia)
Includes: Freiman's Theorem, Lead sentence
Quote, Includes: Freiman's Theorem, Lead sentence
In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate
View the Source
Milliken's tree theorem (Wikipedia)
Includes: Milliken's Tree Theorem, Lead sentence
Quote, Includes: Milliken's Tree Theorem, Lead sentence
In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, o
View the Source
Probabilistic Method (Wikipedia)
Includes: Probabilistic Method, Lead sentence
Quote, Includes: Probabilistic Method, Lead sentence
od is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescr
View the Source
Centered Hexagonal Number (Wikipedia)
Includes: Centered Hexagonal Number, Lead sentence
Quote, Includes: Centered Hexagonal Number, Lead sentence
In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, is a centered figurate number that repr
View the Source
Q-Pochhammer Symbol (Wikipedia)
Includes: Q-Pochhammer Symbol, Lead sentence
Quote, Includes: Q-Pochhammer Symbol, Lead sentence
In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product ( a ; q )
View the Source
Erdos Conjecture on Arithmetic Progressions (Wikipedia)
Includes: Erdos Conjecture on Arithmetic Progressions, Lead sentenceView the Source
Kruskal-Katona theorem (Wikipedia)
Includes: Kruskal-Katona Theorem, Lead sentenceView the Source
Hirsch Conjecture (Wikipedia)
Includes: Hirsch Conjecture, Lead sentence
Quote, Includes: Hirsch Conjecture, Lead sentence
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an
View the Source
Latin Square (Wikipedia)
Includes: Latin Square, Lead sentenceView the Source
Wikipedia: Stirling numbers of the second kind
Includes: Stirling Numbers, Lead sentence
Quote, Includes: Stirling Numbers, Lead sentence
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number o
View the Source
Mirsky's theorem (Wikipedia)
Includes: Mirsky's Theorem, Lead sentence
Quote, Includes: Mirsky's Theorem, Lead sentence
In mathematics, in the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially
View the Source
Polya enumeration theorem (Wikipedia)
Includes: Polya Enumeration Theorem, Lead sentenceView the Source
Steinitz's theorem (Wikipedia)
Includes: Steinitz's Theorem (Convex Polytopes), Lead sentence
Quote, Includes: Steinitz's Theorem (Convex Polytopes), Lead sentence
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by
View the Source
Balinski's theorem (Wikipedia)
Wikimedia FoundationIncludes: Balinski's Theorem, Lead sentence
Quote, Includes: Balinski's Theorem, Lead sentence
In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of thr
View the Source
Sunflower (mathematics), Wikipedia
Includes: Sunflower Lemma, Lead sentenceView the Source
Bertrand's Ballot Theorem (Wikipedia)
Wikimedia FoundationIncludes: Bertrand's Ballot Theorem, Lead sentence
Quote, Includes: Bertrand's Ballot Theorem, Lead sentence
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B re
View the Source
Behrend's theorem (Wikipedia)
Includes: Behrend's Theorem, Lead sentence
Quote, Includes: Behrend's Theorem, Lead sentence
In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to n in which no member of the set i
View the Source
Bruck-Ryser-Chowla theorem (Wikipedia)
Includes: Bruck-Ryser-Chowla Theorem, Lead sentenceView the Source
Dinitz theorem (Wikipedia)
Includes: Dinitz Theorem, Lead sentence
Quote, Includes: Dinitz Theorem, Lead sentence
In combinatorics, the Dinitz theorem, formerly known as the Dinitz conjecture, is a statement about the extension of arrays to par
View the Source
Pál Turán (Wikipedia)
Includes: Pál Turán, Lead paragraph [in-branch]
Quote, Includes: Pál Turán, Lead paragraph [in-branch]
extremal combinatorics
View the Source
Endre Szemerédi (Wikipedia)
Includes: Endre Szemerédi, Lead paragraph [in-branch]
Quote, Includes: Endre Szemerédi, Lead paragraph [in-branch]
combinatorics
View the Source
Issai Schur (Wikipedia)
Includes: Issai Schur, Lead paragraph [in-branch 2]
Quote, Includes: Issai Schur, Lead paragraph [in-branch 2]
combinatorics
View the Source
Nicolaas Govert de Bruijn (Wikipedia)
Includes: Nicolaas Govert de Bruijn, Lead paragraph [in-branch 3]
Quote, Includes: Nicolaas Govert de Bruijn, Lead paragraph [in-branch 3]
combinatorics
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George Pólya (Wikipedia)
Includes: George Pólya, Lead paragraph [in-branch]
Quote, Includes: George Pólya, Lead paragraph [in-branch]
combinatorics
View the Source
Emanuel Sperner (Wikipedia)
Includes: Emanuel Sperner, Lead paragraph [in-branch]
Quote, Includes: Emanuel Sperner, Lead paragraph [in-branch]
Sperner's theorem
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Wikipedia: László Lovász
Includes: László Lovász, Lead
Quote, Includes: László Lovász, Lead
best known for his work in combinatorics
View the Source
Wikipedia: Hao Huang
Includes: Hao Huang, Lead
Quote, Includes: Hao Huang, Lead
Various Problems in Extremal Combinatorics
View the Source
Wikipedia: Percy Alexander MacMahon
Includes: Percy Alexander MacMahon, Lead
Quote, Includes: Percy Alexander MacMahon, Lead
especially noted in connection with the partitions of numbers and enumerative combinatorics
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Wikipedia: William Allen Whitworth
Includes: William Allen Whitworth, Lead
Quote, Includes: William Allen Whitworth, Lead
Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability
View the Source
Wikipedia: Désiré André
Includes: Désiré André, Lead
Quote, Includes: Désiré André, Lead
best known for his work on Catalan numbers and alternating permutations
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Wikipedia: Frank P. Ramsey
Includes: Frank P. Ramsey, Lead
Quote, Includes: Frank P. Ramsey, Lead
So fruitful, in fact, was Ramsey's theorem that today there is an entire branch of mathematics, known as Ramsey theory, which is dedicated to studying similar results.
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Wikipedia: Zsolt Baranyai
Includes: Zsolt Baranyai, Lead
Quote, Includes: Zsolt Baranyai, Lead
Zsolt Baranyai (June 23, 1948 in Budapest, April 6, 1978) was a Hungarian mathematician known for his work in combinatorics.
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Wikipedia: Ben Green (mathematician)
Includes: Ben Green, Lead
Quote, Includes: Ben Green, Lead
Ben Joseph Green FRS (born 27 February 1977) is a British mathematician specialising in combinatorics and number theory.
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Wikipedia: Vera T. Sós
Includes: Vera T. Sós, Lead
Quote, Includes: Vera T. Sós, Lead
Vera Turán Sós (11 September 1930-22 March 2023) was a Hungarian mathematician who specialized in number theory and combinatorics.
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Terence Tao (Wikipedia)
Includes: Terence Tao, Lead paragraph
Quote, Includes: Terence Tao, Lead paragraph
He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory.
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Wikipedia: Steiner system
Includes: Steiner System, Lead paragraph
Quote, Includes: Steiner System, Lead paragraph
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t = 2 or (recently) t ≥ 2.
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Wikipedia: Bell number
Includes: Bell Numbers, Lead paragraph
Quote, Includes: Bell Numbers, Lead paragraph
In combinatorial mathematics, the Bell numbers count the possible partitions of a set.
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Wikipedia: Derangement
Includes: Derangement, Lead paragraph
Quote, Includes: Derangement, Lead paragraph
In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position.
View the Source
Partial Permutation (Wikipedia)
Includes: Partial Permutation, Lead paragraph
Quote, Includes: Partial Permutation, Lead paragraph
In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S.
View the Source
Hook Length Formula (Wikipedia)
Includes: Hook Length Formula, Lead paragraph
Quote, Includes: Hook Length Formula, Lead paragraph
In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram.
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Baranyai's theorem (Wikipedia)
Wikimedia FoundationIncludes: Baranyai's Theorem, Lead paragraph
Quote, Includes: Baranyai's Theorem, Lead paragraph
In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.
View the Source
Aztec diamond (Wikipedia)
Includes: Aztec Diamond Theorem, Lead paragraph
Quote, Includes: Aztec Diamond Theorem, Lead paragraph
In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n.
View the Source
Bondy's theorem (Wikipedia)
Includes: Bondy's Theorem, Lead paragraph
Quote, Includes: Bondy's Theorem, Lead paragraph
It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972.
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Gauss congruence, Wikipedia
Includes: Gauss Congruence, Lead paragraph
Quote, Includes: Gauss Congruence, Lead paragraph
Sequences satisfying Gauss congruence naturally occur in the study of topological dynamics, algebraic number theory and combinatorics.
View the Source
Lambek-Moser theorem (Wikipedia)
Includes: Lambek-Moser Theorem, Lead paragraph
Quote, Includes: Lambek-Moser Theorem, Lead paragraph
The Lambek-Moser theorem belongs to combinatorial number theory.
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Newton's identities, Wikipedia
Includes: Newton's Identities, Lead paragraph
Quote, Includes: Newton's Identities, Lead paragraph
They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
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Lambert W function (Wikipedia)
Includes: Lambert W Function, Lead paragraph
Quote, Includes: Lambert W Function, Lead paragraph
It is useful in combinatorics, for instance, in the enumeration of trees.
View the Source
Wolfram MathWorld: Abel's Binomial Theorem
Wolfram MathWorldIncludes: Abel's Binomial Theorem, Subject classifications
Quote, Includes: Abel's Binomial Theorem, Subject classifications
Discrete Mathematics Combinatorics
View the Source
Fisher's inequality (Wikipedia)
Includes: Fisher's Inequality, Lead section
Quote, Includes: Fisher's Inequality, Lead section
of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population
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Boolean Pythagorean triples problem, Wikipedia
Includes: Boolean Pythagorean Triples Problem, Lead section
Quote, Includes: Boolean Pythagorean Triples Problem, Lead section
The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored
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Catalan Number (Wikipedia)
  • Includes: Catalan Numbers, Lead section
    are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects.
  • Includes: Catalan Number, Lead section
    are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects.
View the Source
Theorem on friends and strangers (Wikipedia)
Includes: Theorem on Friends and Strangers, Lead section
Quote, Includes: Theorem on Friends and Strangers, Lead section
is a mathematical theorem in an area of mathematics called Ramsey theory.
View the Source
Falling and Rising Factorials (Wikipedia)
Includes: Falling and Rising Factorials, Lead section
Quote, Includes: Falling and Rising Factorials, Lead section
In mathematics, the falling factorial (sometimes called the descending factorial, falling
View the Source
Alfréd Rényi, Biography (MacTutor History of Mathematics)
Includes: Alfréd Rényi, Summary
Quote, Includes: Alfréd Rényi, Summary
Alfréd Rényi was a Hungarian mathematician who worked in probability theory, combinatorics, graph theory and number theory.
View the Source
Eugène Charles Catalan, Biography (MacTutor History of Mathematics)
Includes: Eugène Charles Catalan, Summary
Quote, Includes: Eugène Charles Catalan, Summary
Eugène Catalan was a Belgian mathematician who defined the numbers called after him, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals.
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George Szekeres, Biography (MacTutor History of Mathematics)
Includes: George Szekeres, Summary
Quote, Includes: George Szekeres, Summary
George Szekeres was a Hungarian-born mathematician who worked for most of his life in Australia on geometry and combinatorics.
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Jacques Ozanam, Biography (MacTutor History of Mathematics)
Includes: Jacques Ozanam, Biography
Quote, Includes: Jacques Ozanam, Biography
It is certainly for this last work on recreational mathematics that Ozanam will be most remembered.
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