Durfee Square Areas and Associated Partition Identities

Abstract

A partition of an integer $n$ is a representation $n=a_1+a_2+\cdots +a_k$, with integer parts $a_1\ge a_2\ge \cdots \ge a_k\ge 1$. The Durfee square is the largest square of points in the graphical representation of a partition. We consider generating functions for the sum of areas of the Durfee squares for various different classes of partitions of $n$. As a consequence, interesting partition identities are derived. The more general case of Durfee rectangles is also treated, as well as the asymptotic growth of the mean area over all partitions of $n$.