Thermal ellipsoid

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. 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 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:

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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

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