| Περίληψη: | Broadly speaking, combinatorics is the branch of mathematics that deals with different ways of selecting objects from a set or arranging objects. It tries to answer two major kinds of questions, namely, counting questions (how many ways can a selection or arrangement be chosen with a particular set of properties?) and structural questions (does there exist a selection or arrangement of objects with a particular set of properties?). In many cases, the set elements enumeration is a time-consuming and laborious task. This is due to the special form of the elements that make up such sets. Over time, such difficulties became a challenge for mathematics. An enumeration procedure presupposes the understanding of the set’s structure and the form of elements to enumerate. In addition, it is required a more complex analytical approach to the structure of these elements. Usually, the elements of sets under enumeration belongs to clusters of the elements of a certain form. The properties of clusters should be understood and their properties should be defined. Set theory is the basis of combinatorial analysis and the methods for calculation of cardinality. Combinatorics is prerequisite knowledge for understanding Elementary Probability Theory, Elementary Number Theory and Graph Theory.
Combinatorics also includes more complex methods of numbering sets. For example, the indices of sequences of sets are often depicted in power series that thus form generators functions, which can then be analyzed using Mathematical Analysis techniques. Since many enumeration methods involve binomial coefficients, one is not surprised by the appearance of the hypergeometric function. In some cases the enumeration is asymptotic, such as for an example is the estimates for the number of partitions of an integer. In several cases the numeration can be done in a purely synthetic way, using "elementary calculus". Combinatorial methods for determining coefficients are used to determine identities between functions, especially between infinite sums or products such as the well-known ones Ramanujan identities. topic
A topic of Combinatorics, which however does not belong to the area of enumeration techniques, is study of design forms, i.e. sets and their subsets arranged in highly symmetrical or asymmetrical forms. Of these, perhaps the best known are the Latin squares (arrangements of elements in rectangular array with no repetitions in rows or columns). Also known is the Fano level (seven points belonging to seven "lines", each with three points), indicating the relationship with finite geometries. (With proper axiomatic grounding, these tend to take the form of pro-geometries finite fields, although finite planes are more flexible.) Matroids can to be considered as generalized geometries and that is why they are also included in Combinatorial. Let note that graphs are figures consisting of a set of points and a set of connecting edges pairs of points, and as far as Combinatorial is concerned only regular graphs are included, such as complete, the charts Kuratovsky et al.
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