Durfee square

In number theory, a Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of side 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]

For example, 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.

History

Durfee squares are named after William Durfee, a pupil of English mathematician James Joseph Sylvester. In a letter to Arthur Cayley in 1883, Sylvester wrote:^[3]

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

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

See also

• H-index

References

1. ^ Andrews, George E.; Eriksson, Kimmo (2004). Integer Partitions. Cambridge University Press. pp. 76. ISBN 0521600901. 
2. ^ Weisstein, Eric W., "Durfee Square" from MathWorld.
3. ^ Parshall, Karen Hunger (1998). James Joseph Sylvester: life and work in letters. Oxford University Press. pp. 224. ISBN 0198503911.