Stirling Number of the Second Kind of 1

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Theorem

$\displaystyle {1 \brace n} = \delta_{1 n}$

where:

$\displaystyle {1 \brace n}$ denotes a Stirling number of the second kind
$\delta_{1 n}$ is the Kronecker delta.


Proof

\(\displaystyle {1 \brace n}\) \(=\) \(\displaystyle n \times {0 \brace n} + {0 \brace n - 1}\) Definition of Stirling Numbers of the Second Kind
\(\displaystyle \) \(=\) \(\displaystyle n \times \delta_{0 n} + \delta_{0 \paren {n - 1} }\) Definition of Stirling Numbers of the Second Kind
\(\displaystyle \) \(=\) \(\displaystyle \delta_{0 \paren {n - 1} }\) as $n \times \delta_{0 n}$ is either $n \times 0$ or $0 \times 1$ and so always $0$
\(\displaystyle \) \(=\) \(\displaystyle \delta_{1 n}\) $0 = n - 1 \iff 1 = n$

$\blacksquare$


Also see