a07 - CRN 87401 MAD 6206 Combinatorics I Fall 2011...

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MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Assignment 7 Question 1 . (Rotations of the 3-d cube) We talked about coloring the 6 faces of the cube in class so that each face has a different color. We shall color the vertices and edges of the cube in this problem. Assume that colorings are equivalent under rotations. (a) Color the eight vertices of the cube with k colors so that no two vertices have the same color. How many different colorings are there? (b) Color the twelve edges of the cube with k colors so that no two edges have the same color. How many different colorings are there? Question 2 . (Cycle index of the group of rotations on the vertices of the cube) Consider coloring the vertices of the cube, assuming equivalence under rotations. Let G be the corresponding permutation group on the set of vertices of the cube. (a) Find the cycle index of G . (b) Find the number of possible colorings if there are m colors available.
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