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 \lfloor \sqrt{n} \rfloor.

See also

References

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