A partition of n has a Durfee square of side s if s is the largest number such that the partition contains at least s parts with values ≥ s. An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's Ferrers diagram.
The side-length of the Durfee square is known as the rank of the partition.
The Durfee symbol consists of the two partitions represented by the points to the right or below the Durfee square.
The partition 4 + 3 + 3 + 2 + 1 + 1:
has a Durfee square of side 3 (in red) because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4.
Its Durfee symbol consists of the 2 partitions 1 and 3+1.
In a letter to Arthur Cayley in 1883, Sylvester wrote:
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Durfee square.