Durfee square

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.[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 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:[4]

"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 ) = ∑ k = 0 ∞ x k 2 ∏ i = 1 k ( 1 − x i ) 2 {\displaystyle P(x)=\sum _{k=0}^{\infty }{\frac {x^{k^{2}}}{\prod _{i=1}^{k}(1-x^{i})^{2}}}}

where x k 2 {\displaystyle x^{k^{2}}} is the size of the Durfee square, and ( 1 − x i ) 2 {\displaystyle (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).[5]

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 ⌊ n ⌋ {\displaystyle \lfloor {\sqrt {n}}\rfloor } .

See also

  • h-index
  • Jacobi triple product

References

  1. ^ Andrews, George E.; Eriksson, Kimmo (2004). Integer Partitions. Cambridge University Press. p. 76. ISBN 0-521-60090-1.
  2. ^ Canfield, E. Rodney; Corteel, Sylvie; Savage, Carla D. (1998). "Durfee polynomials". Electronic Journal of Combinatorics. 5. Research Paper 32. doi:10.37236/1370. MR 1631751.
  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.
  5. ^ Hardy, Godfrey Harold; Wright, E. M. (1938), An introduction to the theory of numbers. (First ed.), Oxford: Clarendon Press, JFM 64.0093.03, Zbl 0020.29201