# Polya Enumeration Formula with color indifference

Polya Enumeration Formula gives us 6 equivalence classes of 2-colorings of a square. But, in the Polya coloring, the following 2 colorings belong to 2 different equivalence classes:

00
01


and

11
10


(0 and 1 are the 2 colors.)

What is the theory that groups colorings like the above 2 into the same equivalence class? The reason to put the above 2 into the same class would be that we can obtain one from the other by a mapping between the colors (0->1 and 1->0). Is there a formula to obtain the number of equivalence classes of colorings with this constraint?

The PET (if you're referring to the same thing I'm thinking of) is a special case of Burnside's lemma, which still applies here - but the group is slightly larger. Instead of the group $C_4$ of rotations of a square (which I assume is the group of symmetries you care about - it might be $D_4$ instead, so this is ambiguous), you get the direct product of $C_4$ with $C_2$ (the group that permutes the colors). Burnside's lemma still applies to this larger group.