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. 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.Stanley, Richard P. (1999) Enumerative Combinatorics, Volume 2, p. 289. Cambridge University Press. .
The Durfee symbol consists of the two partitions represented by the points to the right or below the Durfee square.
Examples The partition 4 + 3 + 3 + 2 + 1 + 1:
|- style="vertical-align:top; text-align:left;"
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.
History 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:
Generating function
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).
Properties 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 .
See also h-index Jacobi triple product
References