  A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$
that satisfies a number of geometric conditions. Associated to a
$\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers,
called its type, which encodes many of its homological invariants. We
distinguish $\Bbbk$-configurations by counting the number of lines that contain
$d_s$ points of $\mathbb{X}$. In particular, we show that for all integers $m
\gg 0$, the number of such lines is precisely the value of $\Delta
\mathbf{H}_{m\mathbb{X}}(m d_s -1)$. Here, $\Delta \mathbf{H}_{m\mathbb{X}}(-)$
is the first difference of the Hilbert function of the fat points of
multiplicity $m$ supported on $\mathbb{X}$.
