I am trying to solve the following problem:
Find a number of ways to color cells of rectangular $4\times 6$ grid ($24$ cells) with $2$ colors (black and white) up to rotations and reflections given that each edge (border row and column) has at least one white cell.
My plan:
- Apply Polya's enumeration principle to find number of colorings whithout constrains on edge cells. Number of distinct colorings up to symmetry is equal to number of orbits of $G\curvearrowright \operatorname{Map}(X,C)$, which by Burnside's lemma is equal to $\frac{1}{|G|}\sum\limits_{g\in G}|\operatorname{Fix}_{\operatorname{Map}(X,C)}(g)| = \frac{1}{|G|}\sum\limits_{g\in G}|C|^{c(g)}$, where $G$ - symmetry group of rectangle, $g\in G$, $c(g)$ - number of cycles of permutation $g$, $C$ - set of colors.
- Figure out how to solve simplified version of initial problem
Find a number of colorings up to symmetry where first row has no white cells.Using this result we can find number of "incorrect" colorings with no white cells in each subset of edges. - Apply inclusion-exclusion principle and results from steps 1 and 2 to solve the initial problem.
Steps 1 and 3 seem quite clear to me, but I struggle with step 2.
Strategy from step 1 does not seem to be directly applicable to the case with black-colored first row. Indeed set of colorings: $S = \{\phi \in \operatorname{Map}(X,C) : \phi(x) = \text{black } \forall x \in \text{first row} \}$, it is not closed under $G$, $G\curvearrowright S$ is not defined.
Does there exist a common way of solving such colorings-counting problems with pre-assigned colors?