# One to Integer Rising is Integer Factorial

## Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

$1^{\overline n} = n!$

where:

$1^{\overline n}$ denotes the rising factorial
$n!$ denotes the factorial.

## Proof

 $\displaystyle 1^{\overline n}$ $=$ $\displaystyle \dfrac {\paren {1 + n - 1}!} {\paren {1 - 1}!}$ Rising Factorial as Quotient of Factorials $\displaystyle$ $=$ $\displaystyle \dfrac {n!} {0!}$ $\displaystyle$ $=$ $\displaystyle n!$ Factorial of Zero

$\blacksquare$