6.E: Pólya–Redfield Counting (Exercises)

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Oct 1, 2023
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- 6.4: Pólya-Redfield Counting
- Back Matter

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6.2: Groups of Symmetries

Exercise $\PageIndex{2.1}$

Find the 12 permutations of the vertices of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron. Use the labeling below.

Exercise $\PageIndex{2.2}$

Find the 12 permutations of the edges of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron. Use the labeling below.

6.3: Burnside's Theorem

Exercise $\PageIndex{3.1}$

Write the 12 permutations of the vertices of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron in cycle form. Use the labeling below.

Exercise $\PageIndex{3.2}$

Find the number of different colorings of the vertices of a regular tetrahedron with $k$ colors, modulo the rigid motions.

Exercise $\PageIndex{3.3}$

Write the 12 permutations of the edges of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron in cycle form. Use the labeling below.

Exercise $\PageIndex{3.4}$

Find the number of different colorings of the edges of a regular tetrahedron with $k$ colors, modulo the rigid motions.

Exercise $\PageIndex{3.5}$

Find the number of non-isomorphic graphs on 5 vertices "by hand'', that is, using the method of Example 6.3.2.

6.4: Pólya-Redfield Counting

Exercise $\PageIndex{4.1}$

Find the cycle index $P_G$ for the group of permutations of the vertices of a regular tetrahedron induced by the rigid motions. (See Exercise 6.E.3.1 in Section 6.E.)

Exercise $\PageIndex{4.2}$

Using the previous exercise, write out a full inventory of colorings of the vertices of a regular tetrahedron induced by the rigid motions, with three colors, as in Example 6.4.3. You may use Sage or some other computer algebra system.

Exercise $\PageIndex{4.3}$

Find the cycle index $P_G$ for the group of permutations of the edges of $K_5$ . (See Exercise 6.E.3.5 in Section 6.E. Don't use the general formula above.)

Exercise $\PageIndex{4.4}$

Using the previous exercise, write out a full inventory of the graphs on five vertices, as in Example 6.4.4. You may use Sage or some other computer algebra system.