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?



Polya Enumeration Formula with color indifference
There is a paper [A survey of generalizations of Pólya's enumeration theorem](http://alexandria.tue.nl/repository/freearticles/597547.pdf) which discusses a generalization of Polya's theorem involving colors. It gave [Enumerative Combinatorial Problems Concerning Structures](http://alexandria.tue.nl/repository/freearticles/597529.pdf) as a reference for this generalization. I think this is the theorem that you are looking for and and the variant referred to by Yuan in his answer to your question.


