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Development of the Theory About Combinatorial Complexes of Compositiones in Scientific School of C.F. Hindenburg 
Malykh A.E.
Bojko A.S.
To the last decennials of XVIII century methods of combinatorial analysis obtained widespread occurrence, having become important instrument in applied questions of mathematics. With growing of applications the views on the main notions and calculating algorithms were changed. Such rebuilding became necessary because of significant lag the theoretical base of combinatorics. By that time combinatorial symbolism, which would give a high degree of efficiency and usefulness of combinatorial analysis, was not formed. The counting algorithms, which would allow from sets possible complexes to choose namely the one, which reduce to the solution of the problem, were absented. In the last third of XVIII century a new view at the combinatorial analysis and its applications was appeared: the combinatorial doctrine or combinatorics (Combinatorik) and combinatorial analysis (Combinatorische Analysis) began to distinguish. The subject of one of this was the theory of compositions: combinations, permutations, and arrangements with repetitions of elements and without of them, with a given sums and without them; binomial and polynomial theorems and etc. Combinatorial analysis included the applications of combinatorial doctrine to differential and integral calculuses, and later to other mathematical disciplines. Researches of combinatorial school of german mathematicians, whose leader was K.–F. Hindenburg, conducived by this view. Information about him and scientists of his school are provided in article. The formation of literal symbolism is shown. He was the second after G.W. Leibniz, who tried to create the combinatorial doctrine, which would an applicable to other disciplines and, mainly, to mathematical analysis. Creating of combinatorial theory on a strong foundation, which scientists tried to make up in their works was the aim of scientific school. Hindenburg placed his researches in memoirs. Analysis some of them was presented in this article. As a result the estimation of him contribution in development of doctrine about combinatorial types of compositions (combinations, permutations, arrangements as with repetitions of elements and without them) and also its complexions are given. Numerous counting of algorithms and concrete examples were presented of scientist. The additive theory of partitions is shown in his works. Investigations of Hindenburg were realized in quantitative and qualitative directions.
Keywords: combinatorial analysis; combinations, permutations and arrangements with repetitions of elements and without them; well ordered combinations; «useful» and «useless» combinations and permutations; combinatorial complexes of compositions; complexes with a given sums; classes of compositions with repetitions of elements and given sums; quantitative and qualitative construction directions of researches; additive theory partitions of naturals numbers.
Contacts: Email: malych@pspu.ru
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