In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a Durfee square is an attribute of an
integer partition
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...
. 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
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same parti ...
. 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.
Examples
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.
History
Durfee squares are named after
William Pitt Durfee
William Pitt Durfee (5 February 1855 – 17 December 1941) was an American mathematician who introduced Durfee squares. He was a student of James Joseph Sylvester, James Sylvester, and after obtaining his degree in 1883 he became a professor at Ho ...
, a student of English mathematician
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ro ...
. In a letter to
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, C ...
in 1883, Sylvester wrote:
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
The ''h''-index is an author-level metric that measures both the productivity and citation impact of the publications, initially used for an individual scientist or scholar. The ''h''-index correlates with obvious success indicators such as winn ...
References
{{reflist
Number theory
Integer partitions