An expansion for the number of partitions of an integer

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.