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 $\mathbb{ X}$.
Categories: math.AC math.AG