Some results on ordered and unordered factorization of a positive integers

Daniel Yaqubi; Madjid Mirzavaziri

paper

Some results on ordered and unordered factorization of a positive integers

arXiv:1412.0392

Abstract

As a well-known enumerative problem, the number of solutions of the equation $m = m_{1} + ... + m_{k}$ with $m_{1} ⩽ ... ⩽ m_{k}$ in positive integers is $Π (m, k) = \sum_{i = 0}^{k} Π (m - k, i)$ and $Π$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function $μ_{1} (m, k) :=$ the number of solutions of the equation $m = m_{1} ... m_{k}$ with $m_{1} ⩽ ... ⩽ m_{k}$ in positive integers. In particular, using an elementary proof, we give an explicit formula for the cases $k = 1, 2, 3, 4$ .