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NecklacePolynomial

NecklacePolynomial[n, c, Cyclic]
returns a polynomial in the colors in c whose coefficients represent numbers of ways of coloring an n-bead necklace with colors chosen from c, assuming that two colorings are equivalent if one can be obtained from the other by a rotation.
NecklacePolynomial[n, c, Dihedral]
is different in that it considers two colorings equivalent if one can be obtained from the other by a rotation or a flip or both.