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:

:

style="vertical-align:top; text-align:left;"

| File:RedDot.svgFile:RedDot.svgFile:RedDot.svgFile:GrayDot.svg
File:RedDot.svgFile:RedDot.svgFile:RedDot.svg
File:RedDot.svgFile:RedDot.svgFile:RedDot.svg
File:GrayDot.svgFile:GrayDot.svg
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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:

:P(x) = \sum_{k=0}^\infty \frac{x^{k^2}}{\prod_{i=1}^k (1-x^i)^2}

where x^{k^2} is the size of the Durfee square, and (1-x^i)^2 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 \lfloor \sqrt{n} \rfloor.

See also

References