  We study the asymptotic behaviour of the orbit-counting function and a
dynamical Mertens' theorem for the full $G$-shift for a finitely-generated
torsion-free nilpotent group $G$. Using bounds for the M{\"o}bius function on
the lattice of subgroups of finite index and known subgroup growth estimates,
we find a single asymptotic of the shape \[ \sum_{|\tau|\le
N}\frac{1}{e^{h|\tau|}}\sim CN^{\alpha}(\log N)^{\beta} \] where $|\tau|$ is
the cardinality of the finite orbit $\tau$. For the usual orbit-counting
function we find upper and lower bounds together with numerical evidence to
suggest that for actions of non-cyclic groups there is no single asymptotic in
terms of elementary functions.
