Definition:Integer Partition

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Definition

A partition of a (strictly) positive integer $n$ is a way of writing $n$ as a sum of (strictly) positive integers.


Part

In a partition of a (strictly) positive integer, one of the summands in that partition is referred to as a part.


Partition Function

The partition function $p: \Z_{>0} \to \Z_{>0}$ is defined as:

$\forall n \in \Z_{>0}: \map p n =$ the number of partitions of the (strictly) positive integer $n$.


Examples

Partitions of $4$

The integer $4$ can be partitioned as follows:

$4$
$3 + 1$
$2 + 2$
$2 + 1 + 1$
$1 + 1 + 1 + 1$


Partitions of $5$

The integer $5$ can be partitioned as follows:

$5$
$4 + 1$
$3 + 2$
$3 + 1 + 1$
$2 + 2 + 1$
$2 + 1 + 1 + 1$
$1 + 1 + 1 + 1 + 1$


Also known as

An integer partition can also be referred to as a partition if the context is understood.

Some sources may use the word divide to mean to separate into a partition.

This usage is generally not recommended, as it may be conflated with division.


Also see

  • Results about partition theory can be found here.


Historical Note

Integer partitions were originally studied by Leonhard Paul Euler.

Srinivasa Ramanujan also studied them in some depth, and succeeded in proving a considerable number of their properties.


Sources

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