View source for Partition (number theory)

In [[number theory]] and [[combinatorics]], a '''partition''' of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.

For example, 4 can be partitioned in five distinct ways:

<nowiki>
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
</nowiki>

The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1.

A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.

Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general.

== See also ==

* [[Rank of a partition]], a different notion of rank
* [[Crank of a partition]]
* [[Dominance order]]
* [[Factorization]]
* [[Integer factorization]]
* [[Partition of a set]]
* [[Stars and bars (combinatorics)]]
* [[Plane partition]]
* [[Polite number]], defined by partitions into consecutive integers
* [[Multiplicative partition]]
* [[Twelvefold way]]
* [[Ewens's sampling formula]]
* [[Faà di Bruno's formula]]
* [[Multipartition]]
* [[Newton's identities]]
* [[Leibniz's distribution table for integer partitions]]
* [[Smallest-parts function]]
* A Goldbach partition is the partition of an even number into primes (see [[Goldbach's conjecture]])
* [[Kostant's partition function]]

== External links ==

* [https://en.wikipedia.org/wiki/Partition_(number_theory) Partition (number theory)]

[[Category:Combinatorics]]
[[Category:Mathematics]]