In short: I'm considering colorings of colorings. I have a notion of equivalence that involves permuting the objects in the original colorings and a notion of equivalence that involves switching colors in the original colorings together with the coloring of colorings.
Can I calculate the number of inequivalent colorings of colorings using Pólya's enumeration theorem?
If not: Can I - from the number of inequivalent colorings without color-switching - say something about the number of inequivalent colorings with color-switching?
If not: Can I do anything besides using Burnsides Lemma and thereby essentially doubling my work (because the color-switching doubles the number of elements in my symmetry group acting of the colorings of colorings)?
Background/Explanation:
Consider a $3$-D cube and a set of two colors - let's say black and white. I can then color the six faces of that cube. There are $2^6$ ways of coloring, but some of them are equivalent to each other, because they can be transformed into each other by rotating and/or mirroring the cube.
If I call the the symmetry group of the cube $G$, then I can count the essentially different colorings using Pólya's enumeration theorem: $$Z=\frac{1}{|G|} \sum_{g\in G} 2^{\gamma (g)}$$ where $\gamma (g)$ is the number of cycles $g$ splits into when considering it as a permutation of the faces of the cube.
So far so good.
Let's now say my "object" is the set $D$ of all 2-colorings of the graph $P_3$:
$$D=\{⭘\!\!\!-\!\!\!⭘\!\!\!-\!\!\!⭘,⬤\!\!\!-\!\!\!⭘\!\!\!-\!\!\!⭘,⭘\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⭘,⬤\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⭘,⭘\!\!\!-\!\!\!⭘\!\!\!-\!\!\!⬤,⬤\!\!\!-\!\!\!⭘\!\!\!-\!\!\!⬤,⭘\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⬤,⬤\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⬤\} $$
Now consider the set $S$, which are the 2-colorings of the set $D$. If $R$ is the set of colors, then every $s\in S$ is a mapping $D\rightarrow R$.
I still have a notion of equivalence that involves mirroring, i.e. two colorings $s_1,s_2\in S$ are considered equivalent, if $s_1(\sigma_r(d))=s_2(d)$ for every $d\in D$. Where $\sigma_r$ switches the left and right vertices, e.g. $\sigma_r(⬤\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⭘) = ⭘\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⬤$.
I now consider another notion of equivalence: color-switching. This is where I run into problems.
Call two elements $s_1,s_2\in S$ equivalent if $$\overline{s_1(\overline{d})} = s_2(d) \; \text{for every } d\in D$$ where the "overbar" is a slight abuse of notation, denoting both an inversion of color and an inversion of coloring. To give an example for these two cases:
- $\overline{\text{black}} = \text{white}$
- $\overline{⬤\!\!\!-\!\!\!⬤\!\!\!-\!\!\!⭘} = ⭘\!\!\!-\!\!\!⭘\!\!\!-\!\!\!⬤$
My equivalence relation can now not purely be expressed as permutations of $D$, because the operation of color-switching also affects the colors/outputs of $s_1$ and $s_2$.
To accurately calculate the number of inequivalent colorings of colorings, I fell back to Burnside's Lemma and calculated the number of fixed points of:
- The identity
- The mirroring (switching vertices $1$ and $3$)
- The color-switching
- The combination of the mirroring and the color-switching
Dividing the sum of the fixed points by $4$, I get the result of $88$, which is correct, but tedious.
If my set $D$ is, let's say, the set of all colorings of a larger object, with a lot more symmetries, then adding just the color-switch symmetry will essentially double my work and I will have to fall back to Burnside's Lemma.
If I already calculated the number of inequivalent colorings of colorings of a complicated object with a lot of symmetries, can I say something about what happens if I "just" add the color-switching symmetry?
Curiously enough... Pólya's enumeration theorem does give the correct answer in this case, if I consider $s_1$ and $s_2$ equivalent if $s_1(\overline{d})=s_2$ ... which is baffling to me, because this is not the notion of symmetry I want.
It also works when I considered more complex objects, but I have no clue why.
Not all is well though, because it does not work in a simple case where $D$ is the set of colorings of $⭘\!\!\!-\!\!\!⭘$ and the symmetries are "switching of the vertices" and "color-switching" (and their combination). I have no clue why.
Any help would be much appreciated.
(Actual background: I'm trying to calculate the number of different cellular-automata rules. When neighborhoods or the dimension get larger, it quickly becomes a hassle to do by hand.)
