# 6.E: Pólya–Redfield Counting (Exercises)

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## 6.1: Groups of Symmetries

**Ex 6.1.1** 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.

**Ex 6.1.2** 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.

## 6.2: Burnside's Theorem

**Ex 6.2.1** 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.

**Ex 6.2.2** Find the number of different colorings of the vertices of a regular tetrahedron with \(k\) colors, modulo the rigid motions.

**Ex 6.2.3** 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.

**Ex 6.2.4** Find the number of different colorings of the edges of a regular tetrahedron with \(k\) colors, modulo the rigid motions.

**Ex 6.2.5** Find the number of non-isomorphic graphs on 5 vertices "by hand'', that is, using the method of example 6.2.7.

## 6.3: Pólya-Redfield Counting

**Ex 6.3.1** 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 1 in section 6.2.)

**Ex 6.3.2** 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 6.3.5. You may use Sage or some other computer algebra system.

**Ex 6.3.3** Find the cycle index $P_G$ for the group of permutations of the edges of $K_5$. (See exercise 5 in section 6.2. Don't use the general formula above.)

**Ex 6.3.4** Using the previous exercise, write out a full inventory of the graphs on five vertices, as in example 6.3.6. You may use Sage or some other computer algebra system.