  We provide a new way to represent numerical semigroups by showing that the
position of every Ap\'ery set of a numerical semigroup $S$ in the enumeration
of the elements of $S$ is unique, and that $S$ can be re-constructed from this
"position vector." We extend the discussion to more general objects called
numerical sets, and show that there is a one-to-one correspondence between
$m$-tuples of positive integers and the position vectors of numerical sets
closed under addition by $m+1$. We consider the problem of determining which
position vectors correspond to numerical semigroups.
