The Durfee square of a partition is the largest square of nodes that can fit inside the Ferrers Graph of a partition. Durfee squares were named after the American mathematician William Pitt Durfee (5 February 1855 – 17 December 1941), who was a student of James Joseph Sylvester (3 September 1814 – 15 March 1897).

For example, the partition shown below has a Durfee square of size \(3\times 3\).

In a letter to the mathematician Arthur Cayley (16 August 1821 – 26 January 1895), Sylvester remarked that
"Durfee's square is a great invention of the importance of which its author has no conception."

Durfee squares play a part in the combinatorial proof of many partition identities (see the page on Combinatorial Proofs of Partition Identities, for example). Durfee squares also play a part in the analytic proof of some \(q\)-series identities, for example \begin{equation*} 1+ \sum_{k=1}^{\infty} \frac{q^{k^2}}{\prod_{j=1}^{k}(1-q^{j})^2} = \frac{1}{\prod_{j=1}^{\infty}(1-q^{j})}. \end{equation*}

For a fixed non-negative integer \(r\), one can ask for the largest \(k \times (k+r)\) rectangle of nodes that can fit inside the Ferrers Graph of a partition. This the Durfee rectangle of size \(k \times (k+r)\) of the partition.

Durfee rectangles also play a part in proving various combinatorial and analytic identities.

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