Number of Set Partitions by Number of Components/Examples

Examples of Number of Set Partitions by Number of Components

$4$-Sets into $2$ Subsets

A set with $4$ elements $\set {1, 2, 3, 4}$ can be partitioned into $2$ subsets in $\ds {4 \brace 2} = 7$ ways:

Let $T$ be the set with $3$ elements $\set {1, 2, 3}$.

The partition of $T$ into $1$ set is:

$\set {1, 2, 3}$

to which $\set 4$ can be added in one way to achieve:

$\set {1, 2, 3 \mid 4}$

The partitions of $T$ into $2$ sets are:

$\set {1, 2 \mid 3}$

$\set {1, 3 \mid 2}$

$\set {2, 3 \mid 1}$

to which the element $4$ can be added in $2$ ways each:

$\set {1, 2, 4 \mid 3}$

$\set {1, 2 \mid 3, 4}$

$\set {1, 3, 4 \mid 2}$

$\set {1, 3 \mid 2, 4}$

$\set {2, 3, 4 \mid 1}$

$\set {2, 3 \mid 1, 4}$

Thus we have a total of $7$ ways of partitioning a $4$-element set into $2$ components.