# Definition:Integer Partition

## 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.

## 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.