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Co-auteur: John G. Kenneth H. Rosen is an author and mathematician. His interests include discrete mathematics and number theory. He has published several articles in the areas of number theory and mathematical modeling. Bron: Wikipedia.
Toon meer Toon minder. Samenvatting The importance of discrete and combinatorial mathematics continues to increase as the range of applications to computer science, electrical engineering, and the biological sciences grows dramatically. Providing a ready reference for practitioners in the field, the Handbook of Discrete and Combinatorial Mathematics, Second Edition presents additional material on Google's matrix, random graphs, geometric graphs, computational topology, and other key topics.
New chapters highlight essential background information on bioinformatics and computational geometry. Each chapter includes a glossary, definitions, facts, examples, algorithms, major applications, and references. Recensie s No other book covers such a wide range of topics in discrete mathematicsa useful resource. There are glossaries at the beginning of each chapter. Each subsection begins with definitions and goes on to facts, then algorithms boxed and in pseudo -code , lists of mathematical objects, illuminating pictures, examples, lists of applications, and, finally, bibliography and web sources.
A corollary of Hall's theorem is the theorem on the existence of Latin squares, stating that any Latin rectangle of order , , can be extended to a Latin square of order. Another corollary of Hall's theorem: Any non-negative matrix such that. Hall's theorem also implies that the minimum number of rows and columns of a non-negative matrix containing all positive elements is equal to the maximum number of elements that pairwise are not in the same row or in the same column. The extremal property of partially ordered sets, which is analogous to this theorem, is established by the theorem stating that the minimum number of non-intersecting chains is the same as the size of the maximal subset consisting of pairwise-incomparable elements.
The following theorem also bears an extremal character: If for an -set one collects all the combinations of elements and partitions them into non-intersecting classes, then, given an integer , there exists an such that for there is a subset of elements for which all the combinations belong to the same class. The travelling-salesman problem is an extremal problem too; it consists in composing the shortest route visiting towns and returning to the starting point, where the distances between the towns are known.
This problem has applications in the study of transportation networks. Combinatorial problems of an extremal character are considered in the theory of flows in networks and in graph theory. A significant portion of combinatorial analysis consists of enumeration problems. For their solution one either indicates a method of sorting out combinatorial configurations of a given class, or one determines the number of them, or one does both. Typical results of enumeration problems are: The number of permutations of order with cycles is equal to , where is the Stirling number of the first kind, defined by the equation.
A useful device for the solution of enumeration problems is the permanent of a matrix. The permanent of a matrix ; , the elements of which belong to some ring, is defined by the formula. The number of transversals of some family of subsets of a finite set is equal to the permanent of the corresponding incidence matrix. A whole class of problems on the determination of the number of permutations with restricted positions reduces to the calculation of permanents.
For convenience, these problems are sometimes formulated as problems on the arrangement of mutually non-attacking pieces on an chessboard. Connected with the determination of the permanents of certain classes of matrices are variants of the problem of dimers, which arises in the study of the phenomenon of adsorption and consists in the determination of the number of ways of combining the atoms of di-atomic molecules on some surface.
Its solution can also be obtained in terms of Pfaffians cf. Pfaffian , which are certain functions of matrices close to determinants.
The problem of the number of Latin rectangles squares is also connected with the development of effective methods for calculating permanents of certain -matrices. There are a large number of inequalities giving an estimate of the size of the permanent in certain classes of matrices. The determination of the extremal values of the permanent in specific classes of non-negative matrices is of interest.
For a -matrix with given values of the number of ones in the rows one has the estimate. The famous van der Waerden conjecture , that the minimum permanent of a doubly-stochastic matrix of order is equal to was proved, independently, by D. Falikman and G. Egorichev , cf. An important role in the solution of enumeration problems is played by the method of generating functions cf. Generating function. A generating function. According to this definition, generating functions are effectively used for the solution of enumeration problems in parallel with methods of recurrence relations and finite-difference equations.
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In obtaining asymptotic formulas for the generating functions, analytic functions of a real or complex variable are usually employed. In the latter case, the Cauchy integral is applied in finding expressions for the coefficients. In the solution of enumeration problems, an essential role is played by the formalization of the concept of indistinguishability of objects.
Consider the set of configurations. On the set , a group of permutations acts, thus defining an equivalence relation under which , , if there exists an such that for all. To each corresponds a characteristic , where , , are elements of an Abelian group. The characteristic of the configuration is given by the formula.
If is the number of elements with a given value of the characteristic and is the number of inequivalent configurations ,. Symmetric group of. This theorem is based on Burnside's lemma: The number of equivalence classes defined on the set by the permutation group is given by the formula.
In this form it is applied, for example, in the determination of the number of non-isomorphic abstract automata. If , and , where is used as an image times, then the expression. If the numbers contain zeros, ones, etc. Under some specification of the groups and defining the equivalence of configurations , it is possible to give a method of constructing generating functions for the enumeration of the inequivalent configurations. This method, called the general combinatorial scheme, can be subdivided into four particular cases, according as the groups and take values in the identity group or the symmetric groups of corresponding orders.
These particular cases are the models for the majority of the known combinatorial schemes see  , . This models combination schemes of distributing identical objects into different cells, etc. The generating function for the enumeration of inequivalent configurations such that. This models allocation schemes of distributing distinct objects into different cells, etc. This models schemes of distributing identical objects into identical cells, the enumeration of the partitions of natural numbers, etc.
The enumeration of configurations such that. This models schemes of partitioning finite sets into blocks, distributing distinct objects into identical cells, etc.
An important place in combinatorial analysis is taken up by asymptotic methods. They are applied both for the simplification of complex finite expressions for large values of the parameters entering into them, as well as for obtaining approximate formulas in roundabout ways when the exact formulas are unknown.
It is sometimes convenient to formulate a combinatorial problem of an enumerative character as a problem of finding the characteristics of the distribution of some random process.
Discrete and Combinatorial Mathematics, 5th Edition
Such an interpretation makes it possible to apply the well-developed apparatus of probability theory for finding asymptotics or limit theorems. Classical schemes of random allocations of objects in cells are open to a detailed investigation from these points of view; so also are random partitions of sets, the cyclic structure of random permutations, as well as various classes of random graphs, including graphs of mappings see  ,  , . The probabilistic approach is applied in the study of the combinatorial properties of symmetric groups and semi-groups. The limiting distribution of the order of a random element of the symmetric group as has been investigated, as also have the asymptotics of the probability of the generation of random elements of them.