On the number of partitions of $n$ whose product of the summands is at most $n$

TL;DR

This paper derives an explicit formula for counting partitions of an integer n where the product of summands is at most n, also providing insights into multiplicative partitions of n.

Contribution

It introduces a new explicit formula for counting specific partitions and extends the understanding of multiplicative partitions of integers.

Findings

Derived an explicit counting formula for partitions with product constraints

Provided a method to count multiplicative partitions of n

Enhanced combinatorial understanding of partition products

Abstract

We prove an explicit formula to count the partitions of $n$ whose product of the summands is at most $n$. In the process, we also deduce a result to count the multiplicative partitions of $n$.