In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a Durfee square is an attribute of an
integer partition
In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a summation, sum of positive integers. Two sums that differ only in the order of their summands are considered ...
. 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 non-negative 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 ...
. 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 Sylvester, and after obtaining his degree in 1883 he became a professor at Hobart college in 1884 ...
, 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 ...
. In a letter to
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years.
He ...
in 1883, Sylvester wrote:
Generating function
The Durfee square method leads to this
generating function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
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
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 success indicators such as winning t ...
*
Jacobi triple product
In mathematics, the Jacobi triple product is the identity:
:\prod_^\infty
\left( 1 - x^\right)
\left( 1 + x^ y^2\right)
\left( 1 +\frac\right)
= \sum_^\infty x^ y^,
for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y'' ≠ 0. It ...
References
{{reflist
Number theory
Integer partitions