A mathematician who studies combinatorics is called a combinatorialist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, 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.
In the Middle Ages , combinatorics continued to be studied, largely outside of the European civilization. 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. In modern times, the works of 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 explosion 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.
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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. 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 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 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 approach and various tools in analysis and analytic number theory and has connections with statistical mechanics.
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Graphs are basic objects in combinatorics. The questions range from counting e. Although there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects. Design theory is a study of combinatorial designs , which are collections of subsets with certain 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 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.
Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries Euclidean plane , real projective space , etc. This area provides a rich source of examples for design theory.
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It should not be confused with discrete geometry combinatorial geometry. Order theory is the study of partially ordered sets , both finite and infinite. Various examples of partial orders appear in algebra , geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and Boolean algebras. 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 studies extremal questions on set systems. 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. 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. 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?
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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 , simply 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. The study of point cloud data will include work on high dimensional data sets coming from the analysis of images, from neuroscience, from the study of phylogenetic trees, and from shape and feature recognition. The figure above shows a cubical version of the Boy surface an immersed projective plane with a single triple point.
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It is from the paper by A. Ziegler in Experimental Math. The research emphasis at Wesleyan is in pure mathematics and theoretical computer science. One of the distinctive features of our department is the close interaction between the computer science faculty and the mathematics faculty, particularly those in logic and discrete mathematics. Among possible fields of specialization for Ph. The number of graduate students ranges from 16 to 24, with an entering class of four to eight each year. There have always been both male and female students, graduates of small colleges and large universities, and U.
All of the department's recent Ph. Some of these have subsequently taken positions in industry.