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 \(a_1 \geq a_2 \geq \cdots \geq a_k \geq 1\). 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 \(n\).