Integer partition

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. 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 $p(n)$, and a Young diagram 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 $4$.