Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Statistical physics is one of the fundamental theories of Physics, and uses methods of Statistics in solving physical problems Aspects of combinatorics include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have (algebraic combinatorics). Combinatorial enumeration is a subfield of Enumeration that deals with the counting of objects whose symmetries do not exist or if they exist are combinatorial in Combinatorial design theory is the part of combinatorial Mathematics that deals with the existence and construction of systems of finite sets whose In Combinatorics, a branch of Mathematics, a matroid ( or independence structure is a structure that captures the essence of a notion of "independence" Extremal combinatorics is a field of Combinatorics, which is itself a part of Mathematics. Combinatorial optimization is a branch of optimization. Its domain is optimization problems where the set of Feasible solutions is discrete or can be reduced Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Algebraic combinatorics is an area of Mathematics that employs methods of Abstract algebra, notably Group theory and Representation theory, in

Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century (see the page List of combinatorics topics for details of the more recent development of the subject). This is a list of Combinatorics topics. A few decades ago it might have been said that combinatorics is little more than a way to classify poorly-understood problems and One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

There are many combinatorial patterns and theorems related to the structure of combinatoric sets. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements These often focus on a partition or ordered partition of a set. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In combinatorial Mathematics, an ordered partition O of a set S is a Sequence A1, A2 See the List of partition topics for an expanded list of related topics or the List of combinatorics topics for a more general listing. In Mathematics, a partition may be a Partition of a set or an Ordered partition of a set or a Partition of a graph This is a list of Combinatorics topics. A few decades ago it might have been said that combinatorics is little more than a way to classify poorly-understood problems and Some of the more notable results are highlighted below.

An example of a simple combinatorial question is the following: What is the number of possible orderings of a deck of 52 distinct playing cards? The answer is 52! (52 factorial), which is equal to about 8. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k 0658 × 10⁶⁷.

Another example of a more difficult problem: Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person? The answer depends on n. See "Design theory" below.

Combinatorics is used frequently in computer science to obtain estimates on the number of elements of certain sets. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their A mathematician who studies combinatorics is often referred to as a combinatorialist or combinatorist.

Contents 1 History of Combinatorics 1.1 Earliest uses 1.2 Combinatorics in the West 2 Enumerative combinatorics 2.1 Permutations with repetitions 2.2 Permutations without repetitions 2.3 Combinations without repetitions 2.4 Combinations with repetitions 2.5 Fibonacci numbers 3 Structural combinatorics 3.1 Graph theory 3.2 Design theory 3.3 Matroid theory 4 Extremal and probabilistic combinatorics 4.1 Ramsey theory 4.2 Extremal combinatorics 4.3 Probabilistic combinatorics 5 Geometric combinatorics 6 Topological Combinatorics 7 See also 8 References

History of Combinatorics

Earliest uses

The earliest books about combinatorics are from India. A Jainist text, the Bhagabati Sutra, had the first mention of a combinatorics problem; it asked how many ways one could take six tastes one, two, or three tastes at a time. Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. The Bhagabati Sutra was written around 300 BC, and thus was the first book to mention the choice function ^[1]. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial The next ideas of Combinatorics came from Pingala, who was interested in prosody. Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra Specifically, he wanted to know how many ways a six syllable meter could be made from short and long notes. He wrote this problem in the Chanda sutra (also Chandahsutra) in the second century BC ^[2][3]. In addition, he also found the number of meters that had n long notes and k short notes, which is equivalent to finding the binomial coefficients.

The ideas of the Bhagabati were generalized by the Indian mathematician Mahariva in 850 AD, and Pingala's work on prosody was expanded by Bhaskara ^[1][4] and Hemacandra in 1100 AD. Bhaskara was the first known person to find the generalized choice function, although Brahmagupta may have known earlier^[5]. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers^[2].

While India was the first nation to publish results on Combinatorics, there were discoveries by other nations on similar topics. The I Ching ( Wade-Giles) or “Yì Jīng” ( Pinyin) also called “Classic of Changes” or “Book of Changes” is one of the oldest of the The earliest known connection to Combinatorics comes from the Rhind papyrus, problem 79, for the implementation of a geometric series. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish The next milestone is held by the I Ching. The I Ching ( Wade-Giles) or “Yì Jīng” ( Pinyin) also called “Classic of Changes” or “Book of Changes” is one of the oldest of the The book is about what different hexagrams mean, and to do this they needed to know how many possible hexagrams there were. Since each hexagram is a permutation with repetitions of six lines, where each line can be one of two states, solid or dashed, combinatorics yields the result that there are 2⁶ = 64 hexagrams. A monk also may have counted the number of configurations to a game similar to Go around 700 AD^[6]. Although China had relatively few advancements in enumerative combinatorics, they solved a combinatorial design problem, the magic square, around 100 AD^[5]. Combinatorial design theory is the part of combinatorial Mathematics that deals with the existence and construction of systems of finite sets whose In Recreational mathematics, a magic square of order n is an arrangement of n ² numbers usually distinct Integers in a square, such

In Greece, Plutarch wrote that the Xenocrates discovered the number of different syllables possible in the Greek language. Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c This, however, is unlikely because this is one of the few mentions of Combinatorics in Greece. The number they found, also seems too round to be more than a guess. ^[7][6].

Magic squares remained an interest of China, and they began to generalize their original 3×3 square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century^[5]. The Middle East also learned about binomial coefficients from Indian work, and found the connection to polynomial expansion^[8].

Combinatorics in the West

Combinatorics came to Europe in the 13th century through two mathematicians, Leonardo Fibonacci and Jordanus de Nemore. Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci Jordanus de Nemore was a thirteenth-century European Mathematician who wrote treatises on at least 6 different important mathematical subjects the science of weights “algorismi” Fibonacci's Liber Abaci introduced many of the Arabian and Indian ideas to Europe, including that of the Fibonacci numbers^[9][10]. Liber Abaci (1202 also spelled as Liber Abbaci) is an historic book on Arithmetic by Leonardo of Pisa known later by his nickname Fibonacci Jordanus was the first person to arrange the binomial coefficients in a triangle, as he did in proposition 70 of De Arithmetica. This was also done in the Middle East in 1265, and China around 1300^[5]. Today, this triangle is known as Pascal's triangle.

Pascal's contribution to the triangle that bears his name comes from his work on formal proofs about it, in addition to his connection between it and probability^[5]. Together with Leibniz and his ideas about partitions in the 17th century^[11], they are considered the founders of modern combinatorics.

Both Pascal and Leibniz understood that algebra and combinatorics corresponded (aka, binomial expansion was equivalent to the choice function). This was expanded by De Moivre, who found the expansion of a multinomial^[12]. De Moivre also found the formula for derangements using the principle of inclusion-exclusion, a method different from Nikolaus Bernouli, who had found them previously^[5]. He managed to approximate the binomial coefficients and factorial. In Mathematics, the error function (also called the Gauss error function) is a Special function (non- elementary) which occurs in Probability In Mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large Factorials It is named in honour of James Stirling Finally, he found a closed form for the Fibonacci numbers by inventing generating functions^[13][14]. In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n

In the 18th century, Euler worked on problems of combinatorics. In addition to working on several problems of probability which link to combinatorics, he worked on the knights tour, Graeco-Latin square, Eulerian numbers, and others. The Knight's Tour is a mathematical problem involving a knight on a Chessboard. A Graeco-Latin square or Euler square of order n over two sets S and T, each consisting of n symbols is an n × This page discusses a topic in Combinatorics. For "Euler numbers" in number theory see Euler number. He also invented graph theory by solving the Seven Bridges of Königsberg problem, which also led to the formation of topology. The Seven Bridges of Königsberg is a famous historical problem in mathematics Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Finally, he broke ground with partitions by the use of generating functions^[15]. In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n

Enumerative combinatorics

Counting the number of ways that certain patterns can be formed is the central problem of enumerative combinatorics. Two examples of this type of problem are counting combinations and counting permutations (as discussed in the previous section). 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. A mathematical problem is a problem that is amenable to being analyzed and possibly solved with the methods of Mathematics.

The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k For instance, as shown below, the number of different possible orderings of a deck of n cards is f(n) = n!. Often, no closed form is initially available. In these cases, we frequently first derive a recurrence relation, then solve the recurrence to arrive at the desired closed form.

Finally, f(n) may be expressed by a formal power series, called its generating function, which is most commonly either the ordinary generating function

or the exponential generating function

Often, a complicated closed formula yields little insight into the behavior of the counting function as the number of counted objects grows. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n In these cases, a simple asymptotic approximation may be preferable. In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing A function g(n) is an asymptotic approximation to f(n) if as infinity. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In this case, we write .

Once determined, the generating function may allow one to extract all the information given by the previous approaches. In addition, the various natural operations on generating functions such as addition, multiplication, differentiation, etc. , have a combinatorial significance; this allows one to extend results from one combinatorial problem in order to solve others.

Permutations with repetitions

When the order matters, and an object can be chosen more than once, the number of permutations is

where n is the number of objects from which you can choose and r is the number to be chosen. In several fields of Mathematics the term permutation is used with different but closely related meanings

For example, if you have the letters A, B, C, and D and you wish to discover the number of ways to arrange them in three letter patterns (trigrams)

1. order matters (e. A trigram may also refer to Bagua, a philosophical concept in ancient China g. , A-B is different from B-A, both are included as possibilities)
2. an object can be chosen more than once (A-A possible)

you find that there are 4³ or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total.

Permutations without repetitions

When the order matters and each object can be chosen only once, then the number of permutations is

where n is the number of objects from which you can choose, r is the number to be chosen and "!" is the standard symbol meaning factorial. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

For example, if you have five people and are going to choose three out of these, you will have 5!/(5 − 3)! = 60 permutations.

Note that if n = r (meaning the number of chosen elements is equal to the number of elements to choose from; five people and pick all five) then the formula becomes

where 0! = 1.

For example, if you have the same five people and you want to find out how many ways you may arrange them, it would be 5! or 5 × 4 × 3 × 2 × 1 = 120 ways. The reason for this is that you can choose from 5 for the initial slot, then you are left with only 4 to choose from for the second slot etc. Multiplying them together gives the total of 120.

Combinations without repetitions

When the order does not matter and each object can be chosen only once, the number of combinations is the binomial coefficient:

where n is the number of objects from which you can choose and k is the number to be chosen. In combinatorial mathematics, a combination is an un-ordered collection of distinct elements usually of a prescribed size and taken from a given set In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial

For example, if you have ten numbers and wish to choose 5 you would have 10!/(5!(10 − 5)!) = 252 ways to choose. The binomial coefficient is also used to calculate the number of permutations in a lottery.

Combinations with repetitions

Main articles: Multiset#Multiset coefficients and Stars and bars (probability)

When the order does not matter and an object can be chosen more than once, then the number of combinations is

where n is the number of objects from which you can choose and k is the number to be chosen. In Mathematics, a multiset (or bag) is a generalization of a set. In the context of Combinatorial mathematics, stars and bars refers to a trick used to derive certain Combinatorial theorems

For example, if you have ten types of donuts (n) on a menu to choose from and you want three donuts (k) there are (10 + 3 − 1)! / 3!(10 − 1)! = 220 ways to choose (see also multiset). In Mathematics, a multiset (or bag) is a generalization of a set.

Fibonacci numbers

Let f(n) be the number of distinct subsets of the set that do not contain two consecutive integers. When n = 4, we have the sets {}, {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,4}, so f(4) = 8. We count the desired subsets of S(n) by separately counting those subsets that contain element n and those that do not. If a subset contains n, then it does not contain element n − 1. So there are exactly f(n − 2) of the desired subsets that contain element n. The number of subsets that do not contain n is simply f(n − 1). Adding these numbers together, we get the recurrence relation:

where f(1) = 2 and f(2) = 3.

As early as 1202, Leonardo Fibonacci studied these numbers. Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci They are now called Fibonacci numbers; in particular, f(n) is known as the n + 2nd Fibonacci number. In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci Although the recurrence relation allows us to compute every Fibonacci number, the computation is inefficient. However, by using standard techniques to solve recurrence relations, we can reach the closed form solution:

where , the golden ratio. "Difference equation" redirects here It should not be confused with a Differential equation. In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the

In the above example, an asymptotic approximation to f(n) is:

as n becomes large.

Structural combinatorics

Graph theory

Graphs are basic objects in combinatorics. The questions range from counting (e. g. the number of graphs on n vertices with k edges) to structural (e. g. which graphs contain Hamiltonian cycles) to algebraic questions (e. In the mathematical field of Graph theory, a Hamiltonian path is a path in an Undirected graph which visits each vertex exactly once g. given a graph G and two numbers x and y, does the Tutte polynomial T_G(x,y) have a combinatorial interpretation?). In Mathematics, the Tutte polynomial of a Matroid can be regarded as a normalized generalisation of the Chromatic polynomial of a graph. It should be noted that while there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects.

Design theory

A simple result in the block design area of combinatorics is that the problem of forming sets, described in the introduction, has a solution only if n has the form q² + q + 1. In combinatorial Mathematics, a block design (more fully a balanced incomplete block design) is a particular kind of Set system, which has long-standing It is less simple to prove that a solution exists if q is a prime power. In Mathematics, a prime power is a Positive integer power of a Prime number. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for q congruent to 1 or 2 mod 4, then q is a sum of two square numbers. In Mathematics, a square number, sometimes also called a Perfect square, is an Integer that can be written as the square of some other This last result, the Bruck-Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms. The Bruck – Chowla – Ryser theorem is a result on the Combinatorics of Block designs It states that if a ( v, b In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables

When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combinatorics intersect. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics A finite geometry is any geometric system that has only a finite number of points.

Matroid theory

Matroid theory abstracts part of geometry. In Combinatorics, a branch of Mathematics, a matroid ( or independence structure is a structure that captures the essence of a notion of "independence" Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position 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 dependence relation. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors Not only the structure but also enumerative properties belong to matroid theory.

For instance, given a set of n vectors in Euclidean space, what is the largest number of planes they can generate? Answer: the binomial coefficient

Is there a set that generates exactly one less plane? (No, in almost all cases. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial ) These are extremal questions in geometry, as discussed below.

Extremal and probabilistic combinatorics

Many extremal questions deal with set systems. In Mathematics, a hypergraph is a generalization of a graph, where edges can connect any number of vertices. A simple example is the following: what is the largest number of subsets of an n-element set one can have, if no two of the subsets are disjoint? Answer: half the total number of subsets. Proof: Call the n-element set S. Between any subset T and its complement S − T, at most one can be chosen. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation This proves the maximum number of chosen subsets is not greater than half the number of subsets. To show one can attain half the number, pick one element x of S and choose all the subsets that contain x.

A more difficult problem is to characterize the extremal solutions; in this case, to show that no other choice of subsets can attain the maximum number while satisfying the requirement.

Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate. In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing

Ramsey theory

Ramsey theory is a celebrated part of extremal combinatorics. This article provides an introduction For a more detailed and technical article see Ramsey's theorem. It states that any sufficiently large random configuration will contain some sort of order. In Mathematics, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand

Frank P. Ramsey proved that for every integer k there is an integer n, such that every graph on n vertices either contains a clique or an independent set of size k. Frank Plumpton Ramsey ( February 22, 1903 – January 19, 1930) was a British Mathematician who in addition to This is a special case of Ramsey's theorem. This article goes into technical details quite quickly For a slightly gentler introduction see Ramsey theory. For example, given any group of six people, it is always the case that one can find three people out of this group that either all know each other or all do not know each other. The key to the proof in this case is the Pigeonhole Principle: either A knows three of the remaining people, or A does not know three of the remaining people. The pigeonhole principle, also known as Dirichlet's box (or drawer) principle, states that given two Natural numbers n and

Here is a simple proof: Take any one of the six people, call him A. Either A knows three of the remaining people, or A does not know three of the remaining people. Assume the former (the proof is identical if we assume the latter). Let the three people that A knows be B, C, and D. Now either two people from {B,C,D} know each other (in which case we have a group of three people who know each other - these two plus A) or none of B,C,D know each other (in which case we have a group of three people who do not know each other - B,C,D). QED.

Extremal combinatorics

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_n,n. In the Mathematical field of Graph theory, a complete bipartite graph or biclique is a special kind of Bipartite graph where every

Probabilistic combinatorics

Here the questions are of the following type: what is the probability of a certain graph property for a random graph (within a certain class) E. g. 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), simply by observing that the probability of randomly selecting an object with those properties is greater than 0.

Geometric combinatorics

Geometric combinatorics is related to convex and discrete geometry. Convex geometry is the branch of Geometry studying Convex sets mainly in Euclidean space. Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or Combinatorial, either It asks, e. g. how many faces of each dimension can a convex polytope have. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations Metric properties of polytopes play an important role as well, e. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined g. the Cauchy theorem on rigidity of convex polytopes. Cauchy's theorem is a theorem in Geometry, named after Augustin Cauchy. Special polytopes are also considered, such as permutohedron, associahedron and Birkhoff polytope. In Mathematics, the permutohedron of order n is an ( n  &minus 1-dimensional Polytope embedded in an n -dimensional space The Birkhoff polytope B n is the Convex polytope in R N (where N = n ² whose points are

Topological Combinatorics

See main article on Topological combinatorics. The discipline of Combinatorial topology used combinatorial concepts in Topology and in the early 20th century this gradually turned into the field of Algebraic topology

Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Graph theory, graph coloring is a special case of Graph labeling; it is an assignment of labels traditionally called "colors" to elements of a "Cake cutting" redirects here For the wedding tradition see Wedding reception#Wedding_cake. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Operations research, specifically in Decision analysis, a decision tree (or tree diagram is a decision support tool that uses a graph or The necklace problem is a problem in Recreational mathematics, solved in the early 21st century Discrete Morse theory is a combinatorial adaptation of Morse theory.

See also

References

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