  We count algebraic numbers of fixed degree $d$ and fixed (absolute
multiplicative Weil) height $\mathcal{H}$ with precisely $k$ conjugates that
lie inside the open unit disk. We also count the number of values up to
$\mathcal{H}$ that the height assumes on algebraic numbers of degree $d$ with
precisely $k$ conjugates that lie inside the open unit disk. For both counts,
we do not obtain an asymptotic, but only a rough order of growth, which arises
from an asymptotic for the logarithm of the counting function; for the first
count, even this rough order of growth exists only if $k \in \{0,d\}$ or
$\gcd(k,d) = 1$. We therefore study the behaviour in the case where $0 < k < d$
and $\gcd(k,d) > 1$ in more detail. We also count integer polynomials of fixed
degree and fixed Mahler measure with a fixed number of complex zeroes inside
the open unit disk (counted with multiplicities) and study the dynamical
behaviour of the height function.
