  The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone
$V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a
certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the
nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits
in the enhanced nilpotent cone, in terms of representations of the underlying
quiver. This parameterization generalizes naturally to the enhanced cyclic
nilpotent cone. Our parameterizations are different to the previous ones that
have appeared in the literature. Explicit translations between the different
parametrizations are given.
