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. 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.
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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.
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. 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.
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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? 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. Here again probabilistic tools are used to estimate the mixing time. However, with the growth of applications to 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 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 algebra. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, and can be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant.
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.
Contents Set Theory. Such disciplines like topology, combinatorics, partially ordered sets and their associated algebraic structures lattices and Boolean algebras , and metric spaces are increasingly applied in data mining research. This book presents these mathematical foundations of data mining integrated with applications to provide the reader with a comprehensive reference.
Mathematics is presented in a thorough and rigorous manner offering a detailed explanation of each topic, with applications to data mining such as frequent item sets, clustering, decision trees also being discussed. More than exercises are included and they form an integral part of the material. Some of the exercises are in reality supplemental material and their solutions are included.
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The reader is assumed to have a knowledge of elementary analysis. Features and topics: a [ Study of functions and relations a [ Applications are provided throughout a [ Presents graphs and hypergraphs a [ Covers partially ordered sets, lattices and Boolean algebras a [ Finite partially ordered sets a [ Focuses on metric spaces a [ Includes combinatorics a [ Discusses the theory of the Vapnik-Chervonenkis dimension of collections of sets This wide-ranging, thoroughly detailed volume is self-contained and intended for researchers and graduate students, and will prove an invaluable reference tool.
Subject Data mining. Partially ordered sets. Metric spaces. Set theory. Bibliographic information. Publication date Series Advanced information and knowledge processing Reproduction Electronic reproduction.
Show all. From the reviews: "The book is organized into four parts, with a total of 15 chapters. Algebras Pages Simovici, Dan A. Clustering Pages Simovici, Dan A. Combinatorics Pages Simovici, Dan A. Show next xx. Read this book on SpringerLink Author's web-page contains solutions request, errata and addition information.
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