  Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on
evolutions, Harbourne and Huneke give a series of conjectures that relate
symbolic and regular powers of ideals of fat points in $\mathbb P^n$. The
conjectures involve both containment statements and bounds for the initial
degree in which there is a non-zero form in an ideal. Working with initial
degrees, we verify two of these conjectures for special line count
configurations in projective 2-space over an algebraically closed field of
characteristic 0.
