分割 partition

Partition (number theory)

In number theory and combinato rics, a partition of a positive integ er n, also called an integ er

partition, is a way o f writing n as a sum o f positive integ ers. Two sums that differ o nly in the

order of their summands are considered the same partition. (If order matters, the sum becomes a

compo sitio n.) Fo r example, 4 can be partitio ned in five distinct ways:

4

A summand in a partition is also called a part. The number of partitions o f n is given by the

partition functio n p(n). So p(4) = 5. The no tation λ ⊢ n means that λ is a partition of n.

Partitions can be g raphically visualized with Young diagrams or Ferrers diagrams. They o ccur in

a number of branches o f mathematics and physics, including the study o f symmetric

po lynomials, the symmetric group and in group representation theory in general.

Examples

Representations of part it ions

Ferrers diagram

Young diagram

Part it ion f unct ion

Conjugate and self- conjugate partitions

Odd parts and distinct parts

Restricted part size or number of parts

Asymptotics

Partitions in a rectangle and Gaussian binomial coefficients

Rank and Durfee square

Young's lat t ice

See also

Not es

References

about the main diagonal of the square are

conjugate partitions.

Partitions of n with biggest addend k