Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let $s_1$ be the number of edges between vertices of identical coloration, and $s_2$ be the number of edges between vertices of distinct coloration. How many pairs $(s_1, s_2)$ exist?
Also, is it known how many distinct total colorations of $G$, with $k_1$ and $k_2$ colors of type $c_1$ and $c_2$, respectively, exist up to rotational & reflectional symmetry of the graph?
Edit: I need to specify that, in the above problem, $k_1$ and $k_2$ are exactly specified, and I am looking for an exact counting solution as a function of $A$, $B$, and $C$ (i.e. as a function of the edge lengths of the cubic lattice). I apologize for any confusion this may have caused.
Edit 2: Actually I would also be perfectly happy with an exact counting solution where $A$, $B$, and $C$ are exactly specified, and we have an exact counting solution as a function of $k_1$ and $k_2 = (A*B*C) - k_1$.