Partitions and decompositions

A partition of a positive integer $n$ is a way of writing $n$ as a sum of positive integers, where order does not matter. For sets, a partition is a collection of non-empty, disjoint subsets whose union is the original set.

Important properties

• The order of parts in a partition does not matter (e.g., $2+3$ is the same as $3+2$).

• Every element of the set appears in exactly one subset in a set partition.

• The number of partitions of a set with $n$ elements is called the Bell number $B_n$.

• The number of integer partitions of $n$ is denoted $p(n)$.