Total Number of Set Partitions/Examples/3
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Example of Total Number of Set Partitions
Let $S$ be a set whose cardinality is $3$.
Then the number of partitions of $S$ is $5$.
Proof
Let $\map p n$ denote the cardinality of the set of partitions of a set whose cardinality is $n$.
From Total Number of Set Partitions, $\map p n$ is the $n$th Bell number $B_n$.
Thus:
\(\ds \map p 3\) | \(=\) | \(\ds B_3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {3 \brace 1} + {3 \brace 2} + {3 \brace 3}\) | Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 3 + 1\) | Definition of Stirling Numbers of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds 5\) |
$\blacksquare$
Illustration
Let a set $S$ of cardinality $3$ be exemplified by $S = \set {a, b, c}$.
Then the partitions of $S$ are:
- $\set {a, b, c}$
- $\set {a, b \mid c}$
- $\set {a, c \mid b}$
- $\set {b, c \mid a}$
- $\set {a \mid b \mid c}$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $16 \ \text{(ii)}$