You can count everything. Elements of combinatorics

The device of the world assumes the presence of a huge number of various phenomena and objects. At the same time science proves that the basis of this abundance is a set of a certain number of components. Joining in different order, these bricks become the basis for architectural constructions of the world around us. The study of the number of all possible variants of a combination of different components is handled by mathematics, in particular, its section, called combinatorics.

So, discrete values, sets (permutations, combinations, enumerations and placement of elements), as well as relations to them (as an option, partial order) are taken as objects of study. Elements of combinatorics have a close connection with geometry and algebra, they practically became the basis for calculations in probability theory. The widest range of different fields of knowledge can not be imagined without using this field of science. The most in demand this section of mathematics has become in statistical physics, genetics and informatics.

And the beginning of its term "combinatorics" has been taking since 1666. In his work "Discourses on combinatorial art," the mathematician Leibniz laid the foundation for the further development of this section of mathematics.

Very often, using the term "combinatorics", take into account a much wider section of discrete mathematics, which includes, for example, graph theory.

Elements of combinatorics are often presented as models of combinatorial configurations. The placement, permutation, combination, composition and splitting of the number are the main components in which the principles of this section of mathematics were embodied.

Placement is an ordered set of a 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. Combinatorial combination - a set of the number of elements that are included in the data. Kits have differences only in the order of the elements, but they are the same in composition, this is the difference between the combination and 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 used to compile the combinatorial model.

Considering the concept of composition of a number, take it any representation as a sum, ordered from positive integers. But the partition of a number is any representation of it as an unordered sum of positive integers.

Elements of combinatorics have found wide application in the most diverse branches of knowledge. At the same time, this part of mathematics itself underwent such a dramatic development that enabled all the accumulated information baggage in this sphere to be allocated to sections.

Considering the 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 indistinguishability or discernibility of elements, resolution of repetition from identical elements, etc.

To calculate the number of configurations, use the classical 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.

A number of questions in graph theory were added to the structural combinatorics , and the influence of the theory of matroids was traced. Among the sections of the discipline there is also an extreme combinatorial, Ramsey theory, probabilistic, topological, and infinitary combinatorics.