# Definition:Integer Partition

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## Contents

## 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.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(38)$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**partition function**