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Given a complete graph with $n$ unlabeled vertices and $C$ colors, you can color each edge with one of the $C$ colors without any restriction, i.e. two edges connecting to same vertex can have same color. What is the number of such colorings? I tried to first label the vertices and use Polya’s Theorem, but got stuck when I compute the automorphism group of colorings. Seems that it is hard to compute all possible colorings which is invariant under a specific permutation

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