COMBINATORICS AN ORDERED MULTIPLICATIVE DECOMPOSITIONS
DOI:
https://doi.org/10.18413/2687-0932-2020-47-1-126-134Keywords:
ordered multiplicative decomposition, combinatorial problem, canonical decomposition, number of n -profiles of rAbstract
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.
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