A formula for the partition function that "counts"

Abstract

We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then study $D(n,k)$ as a quasipolynomial. We consider the natural polynomial approximation $\tilde{D}(n,k)$ to the quasipolynomial representation of $D(n,k)$. Numerically, the sum $\sum_{1\leq k \leq \sqrt{n}} \tilde{D}(n,k)$ appears to be extremely close to the initial term of the Hardy--Ramanujan--Rademacher convergent series for $p(n)$.