Elementary formulas for integer partitions

TL;DR

This paper presents explicit elementary formulas for counting integer partitions, including the partition function p(n) and the number of partitions into exactly k parts, with detailed summation expressions.

Contribution

It introduces new exact formulas for partition functions, derived through elementary proofs, expanding the computational tools for partition enumeration.

Findings

Derived explicit formulas for p(n) and p(n,k)

Provided summation expressions involving divisors and Möbius function

Formulas are elementary and accessible for computation

Abstract

In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For instance, we shall prove that $$ p(n) = \sum_{d|n} \sum_{k=1}^{d} \sum_{i_0 =1}^{\lfloor d/k \rfloor} \sum_{i_1 =i_0}^{\lfloor\frac{d- i_0}{k-1} \rfloor} \sum_{i_2 =i_1}^{\lfloor\frac{d- i_0 - i_1}{k-2} \rfloor} ... \sum_{i_{k-3}=i_{k-4}}^{\lfloor\frac{n- i_0 - i_1-i_2- ...-i_{k-4}}{3} \rfloor} \sum_{c|(d,i_0,i_1,i_2,...,i_{k-3})} \mu(c) (\lfloor \frac{d-i_0-i_1-i_2- ... i_{k-3}}{2c} \rfloor - \lfloor\frac{i_{k-3}-1}{c} \rfloor).$$ Our proofs are elementary.