Definition
The full scope of combinatorics is not universally agreed upon. According toHistory
Basic combinatorial concepts and enumerative results appeared throughout the Ancient history, ancient world. In the 6th century BCE, Timeline of Indian history, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 26 − 1 possibilities. Ancient Greece, Greek historian Plutarch discusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers. Earlier, in the ''Ostomachion'', Archimedes (3rd century BCE) may have considered the number of configurations of a tiling puzzle, while combinatorial interests possibly were present in lost works by Apollonius of Perga, Apollonius. In the Middle Ages, combinatorics continued to be studied, largely outside of the Culture of Europe, European civilization. The Indian mathematician Mahāvīra (mathematician), Mahāvīra () 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 philosopher and astronomer Rabbi Abraham ibn Ezra () 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 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. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Blaise Pascal, Pascal, Isaac Newton, Newton, Jacob Bernoulli and Leonhard Euler, Euler became foundational in the emerging field. In modern times, the works of James Joseph Sylvester, J.J. Sylvester (late 19th century) and Percy Alexander MacMahon, Percy MacMahon (early 20th century) helped lay the foundation for Enumerative combinatorics, enumerative and algebraic combinatorics. Graph theory also enjoyed an increase of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject. In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field.Approaches and subfields of combinatorics
Enumerative combinatorics
Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. 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. Fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations and Partition of a set, partitions.Analytic combinatorics
Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining Asymptotic analysis, asymptotic formulae.Partition theory
Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the Bijective proof, bijective approach and various tools in analysis and analytic number theory and has connections with statistical mechanics. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches ofGraph theory
Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on ''n'' vertices with ''k'' edges) to existing structures (e.g., Hamiltonian cycles) to algebraic representations (e.g., given a graph ''G'' and two numbers ''x'' and ''y'', does the Tutte polynomial ''T''''G''(''x'',''y'') have a combinatorial interpretation?). Although there are very strong connections between graph theory and combinatorics, they are sometimes thought of as separate subjects. While combinatorial methods apply to many graph theory problems, the two disciplines are generally used to seek solutions to different types of problems.Design theory
Design theory is a study of combinatorial designs, which are collections of subsets with certain Set intersection, intersection properties. Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. 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. The area has further connections to coding theory and geometric combinatorics. Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, Tournament, tournament scheduling, Lottery, lotteries, mathematical chemistry, mathematical biology, Algorithm design, algorithm design and analysis, Computer network, networking, group testing and cryptography.Finite geometry
Finite geometry is the study of Geometry, geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for Combinatorial design, design theory. It should not be confused with discrete geometry (combinatorial geometry).Order theory
Order theory is the study of partially ordered sets, both finite and infinite. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". Various examples of partial orders appear in abstract algebra, algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include Lattice (order), lattices and Boolean algebras.Matroid theory
Matroid theory abstracts part of geometry. 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 independence, linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory. It is now an independent field of study with a number of connections with other parts of combinatorics.Extremal combinatorics
Extremal combinatorics studies how large or how small a collection of finite objects (numbers, Graph (discrete mathematics), graphs, Vector space, vectors, Set (mathematics), sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns Class (set theory), classes of set systems; 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 family#Sperner's theorem, Sperner's theorem, which gave rise to much of extremal set theory. 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 ''Kn,n''. Often it is too hard even to find the extremal answer ''f''(''n'') exactly and one can only give an asymptotic analysis, asymptotic estimate. Ramsey theory is another part of extremal combinatorics. It states that any sufficiently large configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle.Probabilistic combinatorics
In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, 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) by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as ''the'' probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the Markov chain mixing time, mixing time. Often associated with Paul Erdős, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. However, with the growth of applications to analysis of algorithms, analyze algorithms in computer science, as well as classical probability, additive number theory, and probabilistic number theory, the area recently grew to become an independent field of combinatorics.Algebraic combinatorics
Algebraic combinatorics is an area ofCombinatorics on words
Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory, and linguistics. While many applications are new, the classical Chomsky–Schützenberger hierarchy of classes of formal grammars is perhaps the best-known result in the field.Geometric combinatorics
Geometric combinatorics is related to Convex geometry, convex and discrete geometry. It asks, for example, how many faces of each dimension a convex polytope can have. Metric geometry, Metric properties of polytopes play an important role as well, e.g. the Cauchy's theorem (geometry), Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedron, permutohedra, associahedron, associahedra and Birkhoff polytopes. Combinatorial geometry is a historical name for discrete geometry. It includes a number of subareas such as polyhedral combinatorics (the study of Face (geometry), faces of Convex polyhedron, convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. The study of regular polytopes, Archimedean solids, and kissing numbers is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron, associahedron and Birkhoff polytope.Topological combinatorics
Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partition of a set, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.Arithmetic combinatorics
Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory, and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems.Infinitary combinatorics
Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics. Some of the things studied include continuous graphs and Tree (set theory), trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the Continuum (set theory), continuum and combinatorics on successors of singular cardinals. Gian-Carlo Rota used the name ''continuous combinatorics'' to describe geometric probability, since there are many analogies between ''counting'' and ''measure''.Related fields
Combinatorial optimization
Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, Analysis of algorithms, algorithm theory and computational complexity theory.Coding theory
Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of information theory.Discrete and computational geometry
Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.Combinatorics and dynamical systems
Combinatorics and dynamical systems, Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system.Combinatorics and physics
There are increasing interactions between combinatorics and physics, particularly statistical physics. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic polynomial, chromatic and Tutte polynomials on the other hand.See also
* Combinatorial biology * Combinatorial chemistry * Combinatorial data analysis * Combinatorial game theory * Combinatorial group theory * Discrete mathematics * List of combinatorics topics * Phylogenetics * Polynomial method in combinatoricsNotes
References
* Björner, Anders; and Stanley, Richard P.; (2010)External links
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