Fractal form of the partition functions p (n)

Abstract

The fractal family $\left\{p\left(n,k\right), k \in \mathbb{N}\right\}$, describe a rule to calculate the number of partitions obtained by decomposing $n\in \mathbb{N}$, into exactly $k$ parts. In this paper, we will present a novel method for proving that polynomials $\left\{p\left(n,k\right), k \in \mathbb{N} \right\}$ have fractal form. For each class $k$, up to the $LCM\left(1,2,3,\dots,k\right)$, different polynomials of degree $k-1$ are needed to form one quasi-polynomial $p\left(n,k\right)$. All the polynomials (needed for the same class $k$) have all coefficients of the higher degrees ending with the $ \left[\frac{k}{2}\right] $ degree in common. Moreover, we will prove that, for a fixed value of $k$, all the first, second, etc. coefficients of the common part of the fractal family have a general form, showing the vertical connection between the corresponding coefficients of all fractal family $\left\{ p\left(n,k\right), k\in \mathbb{N}\right\} $. Furthermore, for a fixed value of $k$, all the coefficients within the same polynomial have a unique general form, showing the horizontal connection of the coefficients of the polynomial $p\left(n,k\right)$. The partition function is not real a polynomial, but it can be written as a fractal polynomial which can be obtained from the general form of the partition class functions $\left\{p\left(n, k\right)\right\}$. In that case, the partition function for each $n$ uses a different polynomial. We show that all these polynomials can be combined with one single in which each member can be a formula for calculating the total number of partitions of all natural numbers.

Keywords

• partition function
• partition class functions
• fractal polynomial