# 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]} 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.

## 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 3+1.

## History

Durfee squares are named after William 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."

## 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

## References

- ↑ Andrews, George E.; Eriksson, Kimmo (2004).
*Integer Partitions*. Cambridge University Press. p. 76. ISBN 0-521-60090-1. - ↑ Weisstein, Eric W. "Durfee Square".
*MathWorld*. - ↑ Stanley, Richard P. (1999)
*Enumerative Combinatorics*, Volume 2, p. 289. Cambridge University Press. ISBN 0-521-56069-1. - ↑ Parshall, Karen Hunger (1998).
*James Joseph Sylvester: life and work in letters*. Oxford University Press. p. 224. ISBN 0-19-850391-1.