---
tag: 0404R9P9J3
kind: definition
name: Integer partition
msc: [05A17]
defines: 0JV30XN73Z
sources:
  - url: https://en.wikipedia.org/wiki/Integer_partition
    locator: lead section
---

## Definition

Let $n$ be a natural number.
A partition of $n$ is a finite multiset of positive integers whose sum is $n$.
Its elements are the parts of the partition.

Equivalently, a partition of $n$ is a finite sequence $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_k$ of positive integers with $\lambda_1 + \dots + \lambda_k = n$.

## Remarks

The order of the parts does not matter. A sum in which the order matters is a [[composition|0J8ANSYKAC]].
The $5$ partitions of $4$ are $4$, $3 + 1$, $2 + 2$, $2 + 1 + 1$ and $1 + 1 + 1 + 1$.
The number $0$ has exactly one partition, the empty partition.
The number of partitions of $n$ is the [[partition function|001CX45SWQ]] $p(n)$, and a [[Young diagram|0K78D81EF7]] draws a partition.

## Explanations

### L1 human

A partition of $n$ is a way to write $n$ as a sum of positive integers when the order of the summands does not matter.
So $3 + 1$ and $1 + 3$ are the same partition of $4$.