Durfee square

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 ≥ s.^[1] An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's Ferrers diagram.^[2] The side-length of the Durfee square is known as the rank of the partition.^[3]

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 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:^[4]

"Durfee's square is a great invention of the importance of which its author has no conception."

The Durfee square method leads to this generating function for the integer partitions:

${\displaystyle P(x)=\sum _{k=0}^{\infty }{\frac {x^{k^{2}}}{\prod _{i=1}^{k}(1-x^{i})^{2}}}}$

where ${\displaystyle x^{k^{2}}}$ is the size of the Durfee square, and ${\displaystyle (1-x^{i})^{2}}$ 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).^[5]

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 ${\displaystyle \lfloor {\sqrt {n}}\rfloor }$.

• h-index
• Jacobi triple product