Unsigned Stirling Number of the First Kind of Number with Self

Theorem

 * $\ds {n \brack n} = 1$

where $\ds {n \brack n}$ denotes an unsigned Stirling number of the first kind.

Proof
The proof proceeds by induction.

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

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 $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $\ds {k \brack k} = 1$

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

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:
 * $\ds \forall n \in \Z_{\ge 0}: {n \brack n} = 1$

Also see

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


 * Particular Values of Unsigned Stirling Numbers of the First Kind