# 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:

 $\displaystyle {0 \brace 0}$ $=$ $\displaystyle \delta_{0 0}$ Stirling Number of the Second Kind of 0 $\displaystyle$ $=$ $\displaystyle 1$ Definition of Kronecker Delta

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:

 $\displaystyle {k + 1 \brace k + 1}$ $=$ $\displaystyle \paren {k + 1} {k \brace k + 1} + {k \brace k}$ Definition of Stirling Numbers of the Second Kind $\displaystyle$ $=$ $\displaystyle \paren {k + 1} \times 0 + {k \brace k}$ Stirling Number of Number with Greater $\displaystyle$ $=$ $\displaystyle {k \brace k}$ $\displaystyle$ $=$ $\displaystyle 1$ Induction Hypothesis

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

Therefore:

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

$\blacksquare$