The Application of Polya Theorem (2): my notes on an 6 minute Ma Yuchun video.
First Ma Yuchun further analyzes the situation for the problem involving tetrahedral symetry (the chemistry problem) which is explained in my notes at https://www.facebook.com/cj.fearnley/posts/10224496234048995
She explains that we don't need to actually compute the permutations. Instead, we just need to count the number of cycles in each class of permutation and put the sum of the exponents into the formula for Pólya's enumeration theorem which is explained in these notes https://www.facebook.com/cj.fearnley/posts/10224471583672751
So in the tetrahedron calculation, instead of computing the permutations, we just analyze the order of each cycle in each permutation:
For the ±120° rotations with an axis between a vertex and the center of the opposite face, each is a (1)¹(3)¹ permutation (using her notion of {order of the cycle} raised to the power of {number of cycles in the disjoint cycle representation of the permutation}.
And we need to realize that there are 4 vertices with these two rotations, so there are a total of 8 such permutations
In addition, there are 3 axes on the midpoints of opposite edges with (2)² permutations.
There is one identity permutation which is a (1)⁴ in this case (4 vertices).
So there are 12 permutations in the rotation group.
Computing with Pólya's theorem and 4 colors, we get (1/12)[8*4²+3*4²+4⁴]=36.
Next she analyzes the problem of how many ways there are to color a 4-bead necklace with 3 colors?
Rotations, flips, and the fact that some beads may have the same color all complicate the situation.
If the beads look different when turned over, then we do not need to take flipping into account. But if they are symmetrical beads on flipping, then we need to avoid overcounting!
First she computes the asymmetrical bead case:
We have 2 ±90° rotations with 1 4-cycle, so 2 (4)¹.
We have 1 180° rotation with 2 2-cycles, so 1 (2)².
We have 1 identity permutation with 1 1-cycle, so 1 (1)⁴.
We compute: (1/4)(2*3¹+1*3²+1*3⁴)=24.
Now she computes the case of symmetrical beads:
The ±90° and the 180° rotations are the same as is the identity permutation.
Flipping permits one of two reflections across the necklace cutting symmetrically between pairs of beads:
axis 1: there are two such reflections with 2 2-cycles: so 2 (2)².
The other kind of axis cuts two beads in half:
axis 2: there are two such reflections with 1 2-cycle and 2 1-cycles: so 2 (2)¹(1)².
We compute (note: 8 permutations): (1/8)(2*3¹+1*3²+1*3⁴+2*3²+2*3³)=21.
Subtitles: https://www.cjfearnley.com/TsinghuaX60240013.x/week8/8-2-3-en.srt
From the free on-line course: https://www.edx.org/course/combinatorial-mathematics-2
Lecture: Week 8, Polya Theorem, From Burnside to Polya, The Application of Polya Theorem (2)