Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite set structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry,[Björner and Stanley, p. 2] as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.
Definition
The full scope of combinatorics is not universally agreed upon.
According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions.
Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with:
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the enumeration (counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems,
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the existence of such structures that satisfy certain given criteria,
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the construction of these structures, perhaps in many ways, and
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optimization: finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained."
History
Basic combinatorial concepts and enumerative results appeared throughout the
Ancient history. The earliest recorded use of combinatorial techniques comes from problem 79 of the
Rhind papyrus, which dates to the 16th century BC. The problem concerns a certain
geometric series, and has similarities to Fibonacci's problem of counting the number of compositions of 1s and 2s that
Summation to a given total.
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 2
6 − 1 possibilities.
Ancient Greece 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.
[Stanley, Richard P.; "Hipparchus, Plutarch, Schröder, and Hough", American Mathematical Monthly 104 (1997), no. 4, 344–350.] 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.
In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The mathematician Mahāvīra () provided formulae for the number of and ,[. (Translation from 1967 Russian ed.)]
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 on permutations.
During the Renaissance, together with the rest of mathematics and the , combinatorics enjoyed a rebirth. Works of Blaise Pascal, Isaac Newton, Jacob Bernoulli and Leonhard Euler became foundational in the emerging field. In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics. 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.[See Journals in Combinatorics and Graph Theory ] 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 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 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 and various tools in analysis and analytic number theory and has connections with statistical mechanics. Partitions can be graphically visualized with
or
. They occur in a number of branches of
mathematics and
physics, including the study of symmetric polynomials and of the
symmetric group and in group representation theory in general.
Graph 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.
[Sanders, Daniel P.; 2-Digit MSC Comparison ] 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 properties.
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 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, Lottery, mathematical chemistry, mathematical biology, Algorithm design, Computer network, group testing and cryptography.
Finite geometry
Finite geometry is the study of
Geometry 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 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, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include 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 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 (
, graphs,
Vector space, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of
; this is called extremal set theory. For instance, in an
n-element set, what is the largest number of
k-element
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.
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 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 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. The area recently grew to become an independent field of combinatorics.
Algebraic combinatorics
Algebraic combinatorics is an area of
mathematics that employs methods of
abstract algebra, notably
group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in
abstract algebra. Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve
,
, partially ordered sets, or
Finite geometry. On the algebraic side, besides group and representation theory,
lattice theory and commutative algebra are common.
Combinatorics on words
Combinatorics on words deals with
. 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
is perhaps the best-known result in the field.
Geometric combinatorics
Geometric combinatorics is related to
Convex geometry and discrete geometry. It asks, for example, how many faces of each dimension a
convex polytope can have.
Metric geometry properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as
permutohedron,
associahedron 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 faces of convex polyhedra), convex geometry (the study of , in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. The study of , Archimedean solids, and 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, partitions, partially ordered sets,
,
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
.
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
and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum
[Andreas Blass, Combinatorial Cardinal Characteristics of the Continuum, Chapter 6 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010] and combinatorics on successors of singular cardinals.
Gian-Carlo Rota used the name continuous combinatorics
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, 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
and
. 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
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 and
on the other hand.
See also
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Combinatorial biology
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Combinatorial chemistry
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Combinatorial data analysis
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Combinatorial game theory
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Combinatorial group theory
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Discrete mathematics
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List of combinatorics topics
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Phylogenetics
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Polynomial method in combinatorics
Notes
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Björner, Anders; and Stanley, Richard P.; (2010); A Combinatorial Miscellany
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Bóna, Miklós; (2011); A Walk Through Combinatorics (3rd ed.).
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Graham, Ronald L.; Groetschel, Martin; and Lovász, László; eds. (1996); Handbook of Combinatorics, Volumes 1 and 2. Amsterdam, NL, and Cambridge, MA: Elsevier (North-Holland) and MIT Press.
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Lindner, Charles C.; and Rodger, Christopher A.; eds. (1997); Design Theory, CRC-Press. .
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Stanley, Richard P. (1997, 1999); Enumerative Combinatorics, Volumes 1 and 2, Cambridge University Press.
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van Lint, Jacobus H.; and Wilson, Richard M.; (2001); A Course in Combinatorics, 2nd ed., Cambridge University Press.
External links
