# Stirling Number of the Second Kind of 1

## Theorem

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

where:

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

## Proof

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

$\blacksquare$