Durfee square
{{short description|Integer partition attribute, in number theory}}
In number theory, a Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least s parts with values ≥ s.{{Cite book
| last = Andrews
| first = George E.
|author2=Eriksson, Kimmo
| title = Integer Partitions
| publisher = Cambridge University Press
| year = 2004
| pages = 76
| isbn = 0-521-60090-1}} An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's Ferrers diagram.{{cite journal
| last1 = Canfield | first1 = E. Rodney
| last2 = Corteel | first2 = Sylvie | author2-link = Sylvie Corteel
| last3 = Savage | first3 = Carla D. | author3-link = Carla Savage
| doi = 10.37236/1370
| journal = Electronic Journal of Combinatorics
| mr = 1631751
| at = Research Paper 32
| title = Durfee polynomials
| volume = 5
| year = 1998| doi-access = free
}} The side-length of the Durfee square is known as the rank of the partition.Stanley, Richard P. (1999) [http://www-math.mit.edu/~rstan/ec/ Enumerative Combinatorics, Volume 2], p. 289. Cambridge University Press. {{ISBN|0-521-56069-1}}.
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 2+1+1.
History
Durfee squares are named after William Pitt Durfee, a student of English mathematician James Joseph Sylvester. In a letter to Arthur Cayley in 1883, Sylvester wrote:{{Cite book
| last = Parshall | first = Karen Hunger | author-link = Karen Parshall
| title = James Joseph Sylvester: life and work in letters
| publisher = Oxford University Press
| year = 1998
| pages = 224
| isbn = 0-19-850391-1}}
{{quote|"Durfee's square is a great invention of the importance of which its author has no conception."}}
Generating function
The Durfee square method leads to this generating function for the integer partitions:
:
where is the size of the Durfee square, and represents the two sections to the right and below a Durfee square of size k (being two partitions into parts of size at most k, equivalently partitions with at most k parts).{{Citation | last1=Hardy | first1=Godfrey Harold | author1-link=G. H. Hardy | last2=Wright | first2=E. M. | title=An introduction to the theory of numbers. | publisher=Oxford: Clarendon Press | edition=First | year=1938 |jfm = 64.0093.03 | zbl = 0020.29201}}
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 .