Some results on ordered and unordered factorization of a positive integers

TL;DR

This paper explores the enumeration of ordered and unordered factorizations of positive integers, providing recursive and explicit formulas for the multiplicative partition function, extending classical additive partition results.

Contribution

It introduces a recursive formula for the multiplicative partition function and explicit formulas for small cases, advancing understanding of integer factorizations.

Findings

Recursive formula for multiplicative partition function

Explicit formulas for cases k=1,2,3,4

Extension of additive partition enumeration to multiplicative case

Abstract

As a well-known enumerative problem, the number of solutions of the equation $m=m_1+...+m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers is $\Pi(m,k)=\sum_{i=0}^k\Pi(m-k,i)$ and $\Pi$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function $\mu_1(m,k):=$ the number of solutions of the equation $m=m_1... m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers. In particular, using an elementary proof, we give an explicit formula for the cases $k=1,2,3,4$.