Combinatorics an ordered multiplicative decompositions

Author(s):  V.V. Rumbesht, Belgorod National Research University, Belgorod, Russia, rumbesht@bsu.edu.ru,

Е.V. Burdanova, candidate of Sciences, Belgorod National Research University, Belgorod, Russia, burdanova @bsu.edu.ru

Issue:  Volume 47 № 1

Rubric:  Computer simulation history

Annotation:  The article is devoted to finding the solution of combinatorial problems: finding the number of all possible decompositions of an integer r  1 into an ordered set of integer multipliers and finding the number of all possible decompositions of a number r into an ordered set of integer multipliers. Its purpose is to establish an analytical dependence of the number of all possible expansions of a number r in the ordered set of n integer factors on the parameters r and n . Upon reaching the goal, we directly obtain the solution of the first problem, and the solution of the second problem is the sum of the number of all possible expansions of the number r into the ordered set of n integer factors, where n run the natural series. The article shows that the number of such expansions does not depend on the value of the number r , but rather on the set of exponents in its canonical decomposition. We introduce an equivalence relation that allows us to divide the function of the number of all expansions of a number r into an ordered set of n integer factors into classes so that each class corresponds to the function of the number of matrices of a special kind. The following shows how to count the number of such matrices, which resulted in the conclusion of the desired formula.

Keywords:  ordered multiplicative decomposition, combinatorial problem, canonical decomposition, number of n -profiles of r .

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