Universal formulas for the number of partitions
In this paper, a formula that generalizes the total number of partitions of a natural number and the number of all possible decompositions of a certain number of parts can be united in the same formula. An advantage of this formula compared to similar ones is that it is given as a finite sum. Another advantage is that this amount may be expressed as a polynomial whose coefficients can be computed explicitly in an elementary form. The most important advantage of this approach is the fact that it is possible to express the results obtained in the general form, which so far in all similarattempts was not the case. From the general form we will prove as follows: (a) Partition functions, can be written with one fractal polynomial. (b) In partition functions, $p (n)$is the first half of its coefficients with the highest degree which are in common with all unified polynomials that form it. (c) The remaining coefficients vary. The first variablecoefficient can have two values; the next coefficient can have up to six values, etc. The variability of coefficients increases as the degree of polynomials decreases up to a free member whose variability is up to LCM(2, 3, . . . , $n$).
Volume 131, 2021
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