The device of the world implies the existencethe number of various phenomena and objects. In this science proves that the basis of this abundance is a set of a certain number of components. Combined in a different order, these bricks become the basis for architectural constructions of the world around us. The number of all possible combinations of various components is studied by mathematics, in particular, its section, called combinatorics.
So, as objects of study are accepteddiscrete quantities, sets (permutations, combinations, enumeration and placement of elements), as well as relations on them (as an option, partial order). Elements of combinatorics have a close relationship with geometry and algebra, they almost became the basis for calculations in probability theory. A wide range of different areas of knowledge can not be imagined without the use of this field of science. This branch of mathematics has become the most popular in statistical physics, genetics and computer science.
And the beginning of its term "combinatorics" takes from 1666. In his work “Discourses on combinatorial art,” the mathematician Leibniz laid the foundation for the further development of this branch of mathematics.
Very often, using the term "combinatorics", they take into account a much wider section of discrete mathematics, which includes, for example, graph theory.
Elements of combinatorics are often represented ascombinatorial configuration models. Placement, rearrangement, combination, composition and splitting of numbers are the main components in which the principles of this branch of mathematics were embodied.
Placement is an ordered set ofa certain number of components belonging to a certain set, with a clearly defined number of elements. A permutation is a strictly ordered set of a fixed number of elements. Combination combines is a set of a number of elements included in the data. The sets differ only in the order of the elements, but they are identical in composition, this is the difference between the combination and the placement. The number of combinations depends on the size of the set and the number of elements that make up the set, from which the numbers are taken to compose the specified combinatorial model.
Considering the notion of composition of number, takehis every representation as a sum, ordered from positive integers. But splitting a number is any representation of it as an unordered sum of positive integers.
Elements of combinatorics have found wide application inmost different branches of knowledge. At the same time, this part of mathematics itself went through such a striking development that it made it possible to allocate all the accumulated information baggage in this field into sections.
Considering a section of the discipline called“Enumerative combinatorics” (calculating) take into account enumerations or counting the number of all possible configurations (for example, permutations) that are formed from elements of finite sets. It is possible to impose certain restrictions. This includes the indistinguishability or distinguishability of elements, the resolution of the repetition of identical elements, etc.
Чтобы посчитать количество конфигураций, use the classic rules of multiplication and addition. Elements of combinatorics from this section of the discipline are used to solve a wide range of very different tasks.
In the structural combinatorics added a numberquestions of graph theory, traced the influence of the theory of matroids. Extreme combinatorics, Ramsey theory, probabilistic, topological, infinitary combinatorics also stand out among the sections of the discipline.
