# An expansion for the number of partitions of an integer

## TL;DR

This paper derives an asymptotic expansion for the partition function p(n), using probabilistic methods and cumulant expansions to improve understanding of its growth as n becomes large.

## Contribution

It introduces a novel asymptotic expansion for p(n) based on characteristic functions and cumulants, providing explicit formulas and a concise leading factor.

## Key findings

- Derived an asymptotic series in inverse powers of √n
- Provided explicit formulas for cumulants and the leading factor
- Achieved a more accurate approximation for p(n) as n grows large

## Abstract

We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of independent random variables indexed by $t$. The expansion consists of a factor (which is the leading term) times an asymptotic series expansion in inverse powers of a quantity that grows as $\sqrt n$ as $n\to \infty$, and whose coefficients are simple combinatorial expressions. The asymptotic series is obtained by expanding the characteristic functions in terms of the cumulants of the random variables, for which simple and accurate approximations are derived, as well as explicit exact formulae. That computations also give a concise expression for the factor.