Category: Combinatorics

Piecewise syndetic set
In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set is called piecewise syndetic if there exists a finite subset G of such that for every finite sub
Stirling permutation
In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with the additional property that, fo
Semilinear set
No description available.
Kalmanson combinatorial conditions
In mathematics, the Kalmanson combinatorial conditions are a set of conditions on the distance matrix used in determining the solvability of the traveling salesman problem. These conditions apply to a
Lottery mathematics
Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement
Spt function
The spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each partition of a positive integer. It is related to the partition fu
Tucker's lemma
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed n-dimensional ball . Assume T is antipodally s
Extremal combinatorics
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, vector
In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typical
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived fro
Cycle index
In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read of
Trace monoid
In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters t
Discrete Morse theory
Discrete Morse theory is a combinatorial adaptation of Morse theory developed by . The theory has various practical applications in diverse fields of applied mathematics and computer science, such as
Transylvania lottery
In mathematical combinatorics, the Transylvanian lottery is a lottery where three numbers between 1 and 14, inclusive, are picked by the player for any given ticket, and three numbers are chosen rando
Butcher group
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by , is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-
Dickson's lemma
In mathematics, Dickson's lemma states that every set of -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebr
National Resident Matching Program
The National Resident Matching Program (NRMP), also called The Match, is a United States-based private non-profit non-governmental organization created in 1952 to place U.S. medical school students in
Dobiński's formula
In combinatorial mathematics, Dobiński's formula states that the n-th Bell number Bn (i.e., the number of partitions of a set of size n) equals where denotes Euler's number.The formula is named after
Outline of combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
M. Lothaire
M. Lothaire is the pseudonym of a group of mathematicians, many of whom were students of Marcel-Paul Schützenberger. The name is used as the author of several of their joint books about combinatorics
Block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that frequency of the elements satisfies certa
Combinatorial matrix theory
Combinatorial matrix theory is a branch of linear algebra and combinatorics that studies matrices in terms of the patterns of nonzeros and of positive and negative values in their coefficients. Concep
Catalan's constant
In mathematics, Catalan's constant G, is defined by where β is the Dirichlet beta function. Its numerical value is approximately (sequence in the OEIS) G = 0.915965594177219015054603514932384110774…Un
Combinatorial modelling
Combinatorial modelling is the process which lets us identify a suitable mathematical model to reformulate a problem. These combinatorial models will provide, through the combinatorics theory, the ope
Finite geometry
A finite geometry is any geometric system that has only a finite number of points.The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry ba
Combinatorial data analysis
In statistics, combinatorial data analysis (CDA) is the study of data sets where the order in which objects are arranged is important. CDA can be used either to determine how well a given combinatoria
Disjunct matrix
In mathematics, a logical matrix may be described as d-disjunct and/or d-separable. These concepts play a pivotal role in the mathematical area of non-adaptive group testing. In the mathematical liter
In computer science, the Sharp Satisfiability Problem (sometimes called Sharp-SAT or #SAT) is the problem of counting the number of interpretations that satisfies a given Boolean formula, introduced b
Algorithms and Combinatorics
Algorithms and Combinatorics (ISSN 0937-5511) is a book series in mathematics, and particularly in combinatorics and the design and analysis of algorithms. It is published by Springer Science+Business
Infinitary combinatorics
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets.Some of the things studied include continuous graphs and trees, extens
Ky Fan lemma
In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952.
Longest alternating subsequence
In combinatorial mathematics, probability, and computer science, in the longest alternating subsequence problem, one wants to find a subsequence of a given sequence in which the elements are in altern
Meander (mathematics)
In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number
Polynomial method in combinatorics
In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their alge
De Bruijn torus
In combinatorial mathematics, a De Bruijn torus, named after Dutch mathematician Nicolaas Govert de Bruijn, is an array of symbols from an alphabet (often just 0 and 1) that contains every possible ma
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 n ≥ k ≥ 0 and
Baker–Campbell–Hausdorff formula
In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the fo
Partition of a set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defin
Q-Vandermonde identity
In mathematics, in the field of combinatorics, the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that
Blocking set
In geometry, specifically projective geometry, a blocking set is a set of points in a projective plane that every line intersects and that does not contain an entire line. The concept can be generaliz
Borsuk–Ulam theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere
Large set (combinatorics)
In combinatorial mathematics, a large set of positive integers is one such that the infinite sum of the reciprocals diverges. A small set is any subset of the positive integers that is not large; that
Hunt–Szymanski algorithm
In computer science, the Hunt–Szymanski algorithm, also known as Hunt–McIlroy algorithm, is a solution to the longest common subsequence problem. It was one of the first non-heuristic algorithms used
Natural density
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on th
Constant-recursive sequence
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or mor
Sicherman dice
Sicherman dice /ˈsɪkərmən/ are a pair of 6-sided dice with non-standard numbers–one with the sides 1, 2, 2, 3, 3, 4 and the other with the sides 1, 3, 4, 5, 6, 8. They are notable as the only pair of
De Arte Combinatoria
The Dissertatio de arte combinatoria ("Dissertation on the Art of Combinations" or "On the Combinatorial Art") is an early work by Gottfried Leibniz published in 1666 in Leipzig. It is an extended ver
Combinatorial chemistry
Combinatorial chemistry comprises chemical synthetic methods that make it possible to prepare a large number (tens to thousands or even millions) of compounds in a single process. These compound libra
Seriation (statistics)
In combinatorial data analysis, seriation is the process of finding an arrangement of all objects in a set, in a linear order, given a loss function. The main goal is exploratory, to reveal structural
Probabilistic method
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. It works by sh
Rule of product
In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there ar
Combinatorial biology
In biotechnology, combinatorial biology is the creation of a large number of compounds (usually proteins or peptides) through technologies such as phage display. Similar to combinatorial chemistry, co
Laver table
In mathematics, Laver tables (named after Richard Laver, who discovered them towards the end of the 1980s in connection with his works on set theory) are tables of numbers that have certain properties
Arrangement of hyperplanes
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A g
Chip-firing game
The chip-firing game is a one-player game on a graph which was invented around 1983 and since has become an important part of the study of structural combinatorics.
Toothpick sequence
In geometry, the toothpick sequence is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence. The first sta
Erdős conjecture on arithmetic progressions
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additi
Littlewood–Offord problem
In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of of a set of vectors that fall in a given convex set. More formally, if V is a
In mathematics, the hafnian of an adjacency matrix of a graph is the number of perfect matchings in the graph. It was so named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhag
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies o
Sim (pencil game)
Sim is a pencil-and-paper game that is played by two players.
Longest increasing subsequence
In computer science, the longest increasing subsequence problem is to find a subsequence of a given sequence in which the subsequence's elements are in sorted order, lowest to highest, and in which th
Factorial number system
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not f
Generalized arithmetic progression
In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithm
Josephus problem
In computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. A number of people are standing in a circle waiting
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. It is closely related to many o
Rule of division (combinatorics)
In combinatorics, the rule of division is a counting principle. It states that there are n/d ways to do a task if it can be done using a procedure that can be carried out in n ways, and for each way w
Alignments of random points
Alignments of random points in a plane can be demonstrated by statistics to be counter-intuitively easy to find when a large number of random points are marked on a bounded flat surface. This has been
Sperner's lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring
Sunflower (mathematics)
In the mathematical fields of set theory and extremal combinatorics, a sunflower or -system is a collection of sets whose pairwise intersection is constant. This constant intersection is called the ke
Interval order
In mathematics, especially order theory,the interval order for a collection of intervals on the real lineis the partial order corresponding to their left-to-right precedence relation—one interval, I1,
Johnson scheme
In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors X of length ℓ and weight n, such tha
Lehmer code
In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers. It is an instance of a scheme for numbering permu
Extremal Problems For Finite Sets
Extremal Problems For Finite Sets is a mathematics book on the extremal combinatorics of finite sets and families of finite sets. It was written by Péter Frankl and Norihide Tokushige, and published i
Series multisection
In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series then its
Wilf–Zeilberger pair
In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S.
Bhargava factorial
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava a
Multi-index notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of
Virtual knot
In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. He
Erdős–Graham problem
In combinatorial number theory, the Erdős–Graham problem is the problem of proving that, if the set of integers greater than one is partitioned into finitely many subsets, then one of the subsets can
Percolation (from Latin percolare, "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is descri
Perfect ruler
A perfect ruler of length is a ruler with integer markings , for which there exists an integer such that any positive integer is uniquely expressed as the difference for some . This is referred to as
Combinatorial principles
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle
Cyclic sieving
In combinatorial mathematics, cyclic sieving is a phenomenon by which evaluating a generating function for a finite set at roots of unity counts symmetry classes of objects acted on by a cyclic group.
Coins in a fountain
Coins in a fountain is a problem in combinatorial mathematics that involves a generating function. The problem is described below: In how many different number of ways can you arrange n coins with k c
Combinatorial explosion
In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the proble
Transversal (combinatorics)
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
Composition (combinatorics)
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different com
Index of combinatorics articles
No description available.
Musikalisches Würfelspiel
A Musikalisches Würfelspiel (German for "musical dice game") was a system for using dice to randomly generate music from precomposed options. These games were quite popular throughout Western Europe i
Algorithmic Lovász local lemma
In theoretical computer science, the algorithmic Lovász local lemma gives an algorithmic way of constructing objects that obey a system of constraints with limited dependence. Given a finite set of ba
Stable marriage problem
In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of element
Using the Borsuk–Ulam Theorem
Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry is a graduate-level mathematics textbook in topological combinatorics. It describes the use of results in t
Sparse ruler
A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length with marks is a sequence of integers where . The marks and correspond to the end
Weighing matrix
In mathematics, a weighing matrix of order and weight is a matrix with entries from the set such that: Where is the transpose of and is the identity matrix of order . The weight is also called the deg
Dinitz conjecture
In combinatorics, the Dinitz theorem (formerly known as Dinitz conjecture) is a statement about the extension of arrays to partial Latin squares, proposed in 1979 by Jeff Dinitz, and proved in 1994 by
Langford pairing
In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two
Set packing
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems.Suppose one has a finite set S and a list of subsets
Kemnitz's conjecture
In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autum
Road coloring theorem
In graph theory the road coloring theorem, known previously as the road coloring conjecture, deals with synchronized instructions. The issue involves whether by using such instructions, one can reach
Stable roommates problem
In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching f
Cobham's theorem
Cobham's theorem is a theorem in combinatorics on words that has important connections with number theory, notably transcendental numbers, and automata theory. Informally, the theorem gives the condit
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them.
Erdős sumset conjecture
In additive combinatorics, the Erdős sumset conjecture is a conjecture which states that if a subset of the natural numbers has a positive upper density then there are two infinite subsets and of such
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller
Sidon sequence
In number theory, a Sidon sequence is a sequence of natural numbers in which all pairwise sums (for ) are different. Sidon sequences are also called Sidon sets; they are named after the Hungarian math
First passage percolation
First passage percolation is a mathematical method used to describe the paths reachable in a random medium within a given amount of time.
Barycentric-sum problem
Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. T
Li Shanlan identity
In mathematics, in combinatorics, Li Shanlan identity (also called Li Shanlan's summation formula) is a certain combinatorial identity attributed to the nineteenth century Chinese mathematician Li Sha
Difference set
In combinatorics, a difference set is a subset of size of a group of order such that every nonidentity element of can be expressed as a product of elements of in exactly ways. A difference set is said
Incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean
Graph dynamical system
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis
Petkovšek's algorithm
Petkovšek's algorithm (also Hyper) is a computer algebra algorithm that computes a basis of hypergeometric terms solution of its input linear recurrence equation with polynomial coefficients. Equivale
Pascal's rule
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers n and k, where is a binomial coefficient; one i
Singmaster's conjecture
Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on th
Zero-sum problem
In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one
Combinatorics and dynamical systems
The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about n
Largest small octagon
The largest small octagon is the octagon that has the largest area among all convex octagons with unit diameter. The diameter of a polygon is the length of the longest segment joining two of its verti
Sequential dynamical system
Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provi
Combinatorics and physics
Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
Recurrence relation
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the seque
Partial permutation
In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set Sis a bijection between two specified subsets of S. That is, it is defined by two subsets U and V o
Longest repeated substring problem
In computer science, the longest repeated substring problem is the problem of finding the longest substring of a string that occurs at least twice. This problem can be solved in linear time and space
Pigeonhole principle
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, if one has three gloves
Frobenius characteristic map
In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmet
Inversion (discrete mathematics)
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order.
Lindström–Gessel–Viennot lemma
In Mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the number of tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gesse
Random permutation statistics
The statistics of random permutations, such as the cycle structure of a random permutation are of fundamental importance in the analysis of algorithms, especially of sorting algorithms, which operate
All-pairs testing
In computer science, all-pairs testing or pairwise testing is a combinatorial method of software testing that, for each pair of input parameters to a system (typically, a software algorithm), tests al
Shuffle algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of inter
Hook length formula
In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram.It has applications in diverse areas such as represent
Twelvefold way
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, combin
Domino tiling
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a
Chinese monoid
In mathematics, the Chinese monoid is a monoid generated by a totally ordered alphabet with the relations cba = cab = bca for every a ≤ b ≤ c. An algorithm similar to Schensted's algorithm yields char
Rota–Baxter algebra
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathema
Star product
In mathematics, the star product is a method of combining graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian.
Combinatorial number system
In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an int
Symmetric function
In mathematics, a function of variables is symmetric if its value is the same no matter the order of its arguments. For example, a function of two arguments is a symmetric function if and only if for
Murnaghan–Nakayama rule
In group theory, a branch of mathematics, the Murnaghan–Nakayama rule is a combinatorial method to compute irreducible character values of a symmetric group.There are several generalizations of this r
Shortest common supersequence problem
In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subs
In number theory and combinatorics, a multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn a partition. The concept is also found in the theory of Li
Dividing a circle into areas
In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes ca
Linear recurrence with constant coefficients
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equatio
Symbolic method (combinatorics)
In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is
Dittert conjecture
The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis (in combinatorics) concerning the maximum achieved by a particular function of matrices with real, nonnegative entries
Erdős–Szemerédi theorem
The Erdős–Szemerédi theorem in arithmetic combinatorics states that for every finite set of integers, at least one of , the set of pairwise sums or , the set of pairwise products form a significantly
Geometric combinatorics
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra
No-three-in-line problem
The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. This number is at most , because points in a grid would i
Equiangular lines
In geometry, a set of lines is called equiangular if all the lines intersect at a single point, and every pair of lines makes the same angle.
Addition principle
In combinatorics, the addition principle or rule of sum is a basic counting principle. Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways
Lovász local lemma
In probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive (possibly small) probability that none of the event
Aanderaa–Karp–Rosenberg conjecture
In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the numb
Rudin's conjecture
Rudin's conjecture is a mathematical hypothesis (in additive combinatorics and elementary number theory) concerning an upper bound for the number of squares in finite arithmetic progressions. The conj
Independence system
In combinatorial mathematics, an independence system is a pair , where is a finite set and is a collection of subsets of (called the independent sets or feasible sets) with the following properties: 1
Block walking
In combinatorial mathematics, block walking is a method useful in thinking about sums of combinations graphically as "walks" on Pascal's triangle. As the name suggests, block walking problems involve
Longest common subsequence problem
The longest common subsequence (LCS) problem is the problem of finding the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from the longest com
Bent function
In the mathematical field of combinatorics, a bent function is a special type of Boolean function which is maximally non-linear; it is as different as possible from the set of all linear and affine fu
History of combinatorics
The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduc
Incidence matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, t
Markov spectrum
In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation.
Constraint counting
In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc. that are free to be determined, the idea being that in m
Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topolog
♯P-completeness of 01-permanent
The #P-completeness of 01-permanent, sometimes known as Valiant's theorem, is a mathematical proof about the permanent of matrices, considered a seminal result in computational complexity theory. In a
Athanasius Kircher
Athanasius Kircher (2 May 1602 – 27 November 1680) was a German Jesuit scholar and polymath who published around 40 major works, most notably in the fields of comparative religion, geology, and medici
Group testing
In statistics and combinatorial mathematics, group testing is any procedure that breaks up the task of identifying certain objects into tests on groups of items, rather than on individual ones. First
The Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) is a collaboration between Rutgers University, Princeton University, and the research firms AT&T, Bell Labs, Applied Commu
Bernoulli umbra
In Umbral calculus, Bernoulli umbra is an , a formal symbol, defined by the relation , where is the index-lowering operator, also known as evaluation operator and are Bernoulli numbers, called moments
Cameron–Erdős conjecture
In combinatorics, the Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in is The sum of two odd numbers is even, so a set of odd numbers is always s
Topological combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics.
Cyclic order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does n
Orthogonal array
In mathematics, an orthogonal array is a "table" (array) whose entries come from a fixed finite set of symbols (typically, {1,2,...,n}), arranged in such a way that there is an integer t so that for e
Plethystic substitution
Plethystic substitution is a shorthand notation for a common kind of substitution in the algebra of symmetric functions and that of symmetric polynomials. It is essentially basic substitution of varia
Riordan array
A (proper) Riordan array is an infinite lower triangular matrix, , constructed out of two formal power series, of order 0 and of order 1, in such a way that . A Riordan array is an element of the Rior
Star of David theorem
The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972.
Free convolution
Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately
Stars and bars (combinatorics)
In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems.
Isolation lemma
In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist.This is
Gowers' theorem
In mathematics, Gowers' theorem, also known as Gowers' Ramsey theorem and Gowers' FINk theorem, is a theorem in Ramsey theory and combinatorics. It is a Ramsey-theoretic result about functions with fi
Delannoy number
In mathematics, a Delannoy number describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east
3-dimensional matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyp
Counting lemma
The counting lemmas this article discusses are statements in combinatorics and graph theory. The first one extracts information from -regular pairs of subsets of vertices in a graph , in order to guar
Method of distinguished element
In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.
Toida's conjecture
In combinatorial mathematics, Toida's conjecture, due to in 1977, is a refinement of the disproven Ádám's conjecture from 1967.
Lubell–Yamamoto–Meshalkin inequality
In combinatorial mathematics, the Lubell–Yamamoto–Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by , , , and . It i
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
Combinatorial class
In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects