  We give a closed formula for the dimension of all linear systems in
$\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points
lying on a rational normal curve of degree $n$. In particular we give a purely
geometric explanation of the speciality of these linear systems, which is due
to the presence of certain subvarieties in the base locus: linear spans of
points, secant varieties of the rational normal curve or joins between them.
