From Recursions to Asymptotics: On Szekeres' Formula for the Number of Partitions

Abstract

We give a new proof of Szekeres' formula for $P(n,k)$, the number of partitions of the integer $n$ having $k$ or fewer parts. Our proof is based on the recursion formula satisfied by $P(n,k)$ and Taylor's formula. We make no use of the Cauchy integral formula or complex variables. The derivation is presented as a step-by-step procedure, to facilitate its applicationin other situations. As corollaries we obtain the main term of the Hardy-Ramanujan formulas for $p(n)=$ the number of unrestricted partitions of $n$ and for $q(n)=$ the number of partitions of $n$ into distinct parts.