Title

A Formula for the Partition Function That 'Counts

Document Type

Article

Publication Date

6-2016

Publication Title

Annals of Combinatorics

DOI

10.1007/s00026-016-0305-1

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 D~(n,k)D~(n,k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑1≤k≤n√D~(n,k)∑1≤k≤nD~(n,k) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).