Find the pattern inventory for the corner $3$ colorings of a floating pentagon with adjacent corners with different colors.
Workings:
A pentagon has $10$ symmetries.
The $0^\circ, 72^\circ, 144^\circ, 216^\circ$ and $250^\circ$ rotation. And 5 reflections through each corner and out the opposite end of the corner.
(Note: I drawn a picture here of a pentagon labelled $a,b,c,d,e$ clockwise but I'm not sure how to create that here)
The $0^\circ$ rotation is the cycle permutation $(a)(b)(c)(d)(e)$
The $72^\circ$ rotation is the cycle permutation $(abcde)$
The $144^\circ$ rotation is the cycle permutation $(acebd)$
The $216^\circ$ rotation is the cycle permutation $(adbec)$
The $288^\circ$ rotation is the cycle permuation is $(aedcb)$
The $a$ reflection is the cycle permutation $(a)(b)(e)(cd)$
The $b$ reflection is the cycle permutation $(b)(ac)(de)$
The $c$ reflection is the cycle permutation $(c)(ae)(bd)$
The $d$ reflection is the cycle permutation $(d)(ab)(ce)$
The $e$ reflection is the cycle permutation $(e)(ad)(bc)$
The cycle structure of $0$ is $x_1^5$
The cycle structure of the rest of the rotations is $x_5$
The cycle structure of the reflections is $x_1x_2^2$
Summing up gives:
$P_G = \frac{1}{10} (x_1^8 + 4x_5 + 5x_1x_2^2)$
So for 3 colors gives
$$\frac{1}{10}[(b+w+r)^8+4(b^5+w^5+r^5) + 5(b+w+r)(b^2+w^2+r^2)^2]$$
I'm not sure if I did this right. Especially the adjacent colors with different colors part.
Any help will be appreciated.
Best Answer
You've likely already discovered the answer, but I'm going to post it to help out with whoever else might stumble across this page.
Lets use the colors red (r), green (g) and blue (b). We see that there are three ways for us to label the pentagon (up-to-symmetry): (r,g,b,g,b), (r,g,b,r,g) and (r,g,b,r,b). Since there is only one way to draw each of these pentagrams, then the pattern inventory becomes \begin{align*} rg^2b^2+r^2g^2b+r^2gb^2. \end{align*}