Stirling Number of the Second Kind of 1

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \left\{ {1 \atop n}\right\} = \delta_{1 n}$

where:

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


Proof

\(\displaystyle \left\{ {1 \atop n}\right\}\) \(=\) \(\displaystyle n \times \left\{ {0 \atop n}\right\} + \left\{ {0 \atop {n - 1} }\right\}\) Definition of Stirling Numbers of the Second Kind
\(\displaystyle \) \(=\) \(\displaystyle n \times \delta_{0 n} + \delta_{0 \left({n - 1}\right)}\) Definition of Stirling Numbers of the Second Kind
\(\displaystyle \) \(=\) \(\displaystyle \delta_{0 \left({n - 1}\right)}\) 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