# Unordered Factorizations with $k$ Parts

## TL;DR

This paper introduces new formulas for counting unordered factorizations of integers into k parts, revealing connections with partition theory, Bell numbers, and Stirling numbers, and providing recursive methods for these counts.

## Contribution

It presents novel formulas and recursive methods for enumerating unordered factorizations with k parts, linking them to partition-related combinatorial numbers.

## Key findings

- Derived formulas for unordered factorizations with k parts
- Established relations between partitions, Bell numbers, and Stirling numbers
- Provided recursive formulas for factorizations and partitions with distinct parts

## Abstract

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$, where the parts have a specific number of colors. As a consequence, some new relations between partitions, Bell numbers and Stirling number of the second kind are derived. We also derive a recursive formula for the number of unordered factorizations with $k$ different parts and a simple recursive formula for the number of partitions with $k$ different parts.