Necklace Counting

Distinct arrangements of n beads in k colors under rotation (and reflection)

Necklaces(n,k) = (1/n) Σ_{d|n} φ(d)·k^(n/d)  |  Bracelets: also factor reflections

Necklaces count rotationally distinct colorings. Bracelets also identify reflections (flip). The Burnside/Pólya formula uses Euler's totient φ to count orbits under the cyclic group.