Stirling Number of the Second Kind of Number with Self

Theorem

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

where $\displaystyle {n \brace n}$ denotes a Stirling number of the second kind.

Proof
The proof proceeds by induction.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $\displaystyle {n \brace n} = 1$

Basis for the Induction
$\map P 0$ is the case:

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $\displaystyle {k \brace k} = 1$

from which it is to be shown that:
 * $\displaystyle {k + 1 \brace k + 1} = 1$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \Z_{\ge 0}: {n \brace n} = 1$

Also see

 * Unsigned Stirling Number of the First Kind of Number with Self
 * Signed Stirling Number of the First Kind of Number with Self


 * Particular Values of Stirling Numbers of the Second Kind