Combinatorics in the context of Counting

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of C (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal:

• One variation is that there is a bijection f from the transversal to C such that x is an element of f(x) for each x in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR).
• The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of C. In this situation, the members of the system of representatives are not necessarily distinct.

In computer science, computing transversals is useful in several application domains, with the input family of sets often being described as a hypergraph.

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.

In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. For example, of three gloves, at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into. This seemingly obvious statement, a type of counting argument, can be used to demonstrate possibly unexpected results. For example, given that the population of London is more than one unit greater than the maximum number of hairs that can be on a human head, the principle requires that there must be at least two people in London who have the same number of hairs on their heads.

Although the pigeonhole principle appears as early as 1622 in a book by Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip ("drawer principle" or "shelf principle").

In mathematics, the factorial of a non-negative integer ${\displaystyle n}$, denoted by ${\displaystyle n!}$, is the product of all positive integers less than or equal to ${\displaystyle n}$. The factorial of ${\displaystyle n}$ also equals the product of ${\displaystyle n}$ with the next smaller factorial:$${\displaystyle {\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\&=n\times (n-1)!\\\end{aligned}}}$$For example,$${\displaystyle 5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.}$$The value of 0! is 1, according to the convention for an empty product.

Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of ${\displaystyle n}$ distinct objects: there are ${\displaystyle n!}$. In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science.

Sudoku (/suːˈdoʊkuː, -ˈdɒk-, sə-/; Japanese: 数独, romanized: sūdoku, lit. 'digit-single'; originally called Number Place) is a logic-based, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row, and each of the nine 3 × 3 subgrids that compose the grid (also called "boxes", "blocks", or "regions") contains all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which, for a well-posed puzzle, has a single solution.

French newspapers featured similar puzzles in the 19th century, and the modern form of the puzzle first appeared in 1979 puzzle books by Dell Magazines under the name Number Place. However, the puzzle type only began to gain widespread popularity in 1986 when it was published by the Japanese puzzle company Nikoli under the name Sudoku, meaning "single number". In newspapers outside of Japan, it first appeared in The Conway Daily Sun (New Hampshire) in September 2004, and then The Times (London) in November 2004, both of which were thanks to the efforts of the Hong Kong judge Wayne Gould, who devised a computer program to rapidly produce unique puzzles.

In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook–Levin theorem) to show that there is a polynomial time many-one reduction from the boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete. This was one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable, and it drove interest in the study of NP-completeness and the P versus NP problem.

Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.

Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions.

Much of extremal combinatorics concerns classes of sets; this is called extremal set theory. For instance, in an n-element set, what is the largest number of k-element subsets 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's theorem, which gave rise to much of extremal set theory.

In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s.

The Bell numbers are denoted ${\displaystyle B_{n}}$, where ${\displaystyle n}$ is an integer greater than or equal to zero. Starting with ${\displaystyle B_{0}=B_{1}=1}$, the first few Bell numbers are

In mathematics, a permutation of a 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 of the first meaning is the six permutations (orderings) of the set {1, 2, 3}: written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory.

In mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying purely geometric properties such as the position of vertices.

A geometric polytope is said to be a realization of an abstract polytope in some real n-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.

In algebraic geometry, a toric variety or torus embedding is a kind of algebraic variety that contains an algebraic torus whose group action extends to the whole variety. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special but still quite general class of toric varieties, this information is also encoded in a convex polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.

Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets S_i indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in S_n for each n. 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. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.

The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of n cards is f(n) = n!. The problem of finding a closed formula is known as algebraic enumeration, and frequently involves deriving a recurrence relation or generating function and using this to arrive at the desired closed form.

In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.