# Cycle index for rotation

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• June 3rd 2013, 07:00 PM
Yuuki
Cycle index for rotation
Consider a square where there are points on the four corners and the four midpoints. Find $\hat{C}(r,b)$ using the cycle index for the group D4.

The identity has 8 cycles of length 1; the three rotations have 2 cycles of length 4; the reflections have 2 cycles of length 1 and 3 cycles of length 2.
Hence, the cycle index is:
$(z_{1}^{8}+3z_{4}^{2}+4z_{1}^{2}z_{2}^{3})/8$
So $\hat{C}(r,b)=((r+b)^8+3(r^4+b^4)^2+4(r+b)^2(r^2+b^ 2)^3)/8$
But when I expanded this using Wolfram Alpha, I got fractions for some of the coefficients.
I can't seem to find what I did wrong, so I need help finding out.
Thanks in advance.
• June 4th 2013, 07:12 PM
johng
Re: Cycle index for rotation
Hi,
What are "C hat", b and r? I'm totally ignorant about the usage of the cycle index, but I do know about the dihedral group. Rotation of the square through 180 degrees does not yield a product of two 4 cycles, but a permutation which is the product of four 2 cycles. See the attachment for the computation of the cycle index.

Attachment 28527
• June 7th 2013, 05:27 AM
Yuuki
Re: Cycle index for rotation
The C hat computes the number of possible colorings of corners with different distributions of red and blue.
But your answer showed that I made a mistake in my calculation; thank you.