Periodic Necklaces and the Mobius Switch: Counting Primitive Colorings

We dive into the rebel side of necklace counting: aperiodic (period-n) colorings that stay unique under every nontrivial rotation. Using Burnside’s lemma and the Möbius function, we derive the primitive-necklace formula a(n,k) = (1/n) ∑_{d|n} μ(d) k^{n/d} for counting these primitive patterns. We’ll unpack what μ does, work through a quick example (n = 6, k = 2), and connect this to the broader OEIS landscape, setting the stage for the bracelet story when reflections come into play in the next episode.

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