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2013-04-02T18:47:34Z

146.186.10.12:

In [[number theory]], a '''Durfee square''' is an attribute of an [[Partition (number theory)|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''.<ref>{{Cite book
| last = Andrews
| first = George E.
| coauthors = Eriksson, Kimmo
| title = Integer Partitions
| publisher = Cambridge University Press
| year = 2004
| pages = 76
| isbn = 0-521-60090-1}}</ref> An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's [[Ferrers diagram]].<ref>{{MathWorld |urlname=DurfeeSquare |title=Durfee Square }}</ref>

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:

:{|
|- style="vertical-align:top; text-align:left;"
| [[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]] [[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]] [[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]] [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]] [[File:GrayDot.svg|16px|*]] [[File:GrayDot.svg|16px|*]]
|}

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:<ref>{{Cite book
| last = Parshall
| first = Karen Hunger
| title = James Joseph Sylvester: life and work in letters
| publisher = Oxford University Press
| year = 1998
| pages = 224
| isbn = 0-19-850391-1}}</ref>

{{quote|"''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 <math>\lfloor \sqrt{n} \rfloor</math>.

==See also==
*[[H-index]]

==References==
{{reflist}}

[[Category:Number theory]]

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