In number theory, a Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least s parts with values âÂÂ¥ 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 âÂÂ¥ 3, but does not contain 4 parts that are âÂÂ¥ 4. Its Durfee symbol consists of the 2 partitions 1 and 2+1+1.
Durfee squares are named after William Pitt Durfee, a student of English mathematician James Joseph Sylvester. In a letter to Arthur Cayley in 1883, Sylvester wrote:
The Durfee square method leads to this generating function for the integer partitions:
where is the size of the Durfee square, and represents the two sections to the right and below a Durfee square of size k (being two partitions into parts of size at most k, equivalently partitions with at most k parts).
It is clear from the visual definition that the Durfee square of a partition and its conjugate partition have the same size. The partitions of an integer n contain Durfee squares with sides up to and including .