# Combinatorics

combinatorialcombinatorial mathematicscombinatorial analysiscombinatoriccombinatorialistCombinatorial Theoryprobabilistic combinatoricscombinationscombinatorial identitiescombinatorial objects
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.wikipedia
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### H. J. Ryser

Herbert John RyserHerbert J. RyserHerbert Ryser
Herbert John Ryser (July 28, 1923 – July 12, 1985) was a professor of mathematics, widely regarded as one of the major figures in combinatorics in the 20th century.

### Finite set

finitefinitelyfinite sets
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
:Finite sets are particularly important in combinatorics, the mathematical study of counting.

### Discrete mathematics

discretediscrete mathdiscrete structure
Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.
Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.

### Binomial coefficient

binomial coefficientschoose(generalized) binomial coefficient
1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321.
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics.

### Probability theory

theory of probabilityprobabilityprobability theorist
Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory.
Initially, probability theory mainly considered events, and its methods were mainly combinatorial.

### Enumerative combinatorics

combinatorial enumerationenumerativecombinatorics
In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics.
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.

### Pascal's triangle

Pascal triangleJia Xian triangleKhayyam triangle
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.
Centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greeks' study of figurate numbers.

### Algebraic combinatorics

algebraicCombinatorial-Algebraic
In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics.
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.

### Twelvefold way

permutations and combinationscombinationcombinations
The twelvefold way provides a unified framework for counting permutations, combinations and partitions.
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.

### James Joseph Sylvester

J. J. SylvesterSylvesterJames Sylvester
In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics.
He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics.

### Symbolic method (combinatorics)

analytic combinatoricsAsymptotic combinatoricssymbolic combinatorics
Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory.
In combinatorics, especially in analytic combinatorics, the symbolic method is a technique for counting combinatorial objects.

### Permutation

permutationscycle notationpermuted
The twelvefold way provides a unified framework for counting permutations, combinations and partitions. 850) provided formulae for the number of permutations and combinations, and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.
The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.

### Partition (number theory)

partitionpartitionsinteger partition
Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials.
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.

### Mathematics

mathematicalmathmathematician
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
Combinatorics studies ways of enumerating the number of objects that fit a given structure.

### Q-Pochhammer symbol

q-seriesq''-Pochhammer symbolq''-factorial
Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials.
In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a q-analog of the Pochhammer symbol.

### Bijective proof

bijectivebijectionbijective approach
It incorporates the bijective approach and various tools in analysis and analytic number theory and has connections with statistical mechanics.
In combinatorics, bijective proof is a proof technique that finds a bijective function (that is, a one-to-one and onto function)

### Combinatorial design

combinatorial design theoryDesign theory(''n'',''r'',''k'',''r'')-lotto design
Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties.
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.

### Block design

balanced incomplete block designIncomplete block designPaley biplane
Block designs are combinatorial designs of a special type.
In combinatorial mathematics, a block design is a set together with a family of subsets (repeated subsets are allowed at times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular application.

### Steiner system

Steiner triple systemWitt designSteiner triple systems
The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups.
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.

### Kirkman's schoolgirl problem

15 schoolgirl problemKirkman triple system
This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850.
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48).

### Leon Mirsky

Mirsky, LeonMirsky, L.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained."
In the mid 1960s, Mirsky's research focus shifted again, to combinatorics, after using Hall's marriage theorem in connection with his work on doubly stochastic matrices.

### Geometry

geometricgeometricalgeometries
Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas.
It shares many methods and principles with combinatorics.

### Pigeonhole principle

pigeonholepigeon hole principlecannot exceed
It is an advanced generalization of the pigeonhole principle.
It is an example of a counting argument.

### Probabilistic method

probabilistic combinatoricsprobabilistic argumentProbabilistic methods
This approach (often referred to as the probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory.
The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object.

### Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
In the Ostomachion, Archimedes (3rd century BCE) considers a tiling puzzle.