Burnside's Lemma and Polya Counting

If you naively count the ways to color the four beads of a bracelet with two colors, you get $2^4 = 16$ colorings. But a bracelet can be rotated freely, so a coloring and its rotated version aren't really "different" bracelets — they're the same object viewed differently. Just dividing 16 by the number of rotations (4) doesn't work in general, because some colorings are symmetric under rotation and get overcounted less than others. Burnside's Lemma gives the correct way to count "distinct up to symmetry" objects.

The lemma states that the number of genuinely distinct arrangements equals the average, over every symmetry operation in the group (including "do nothing"), of the number of arrangements left completely unchanged by that operation:

$$\#\text{distinct} = \frac{1}{|G|} \sum_{g \in G} \#\text{fixed}(g) .$$

In plain English: instead of one flat division, you check each possible rotation or reflection separately, count how many colorings look identical before and after that specific move, and average those counts across all the moves in the symmetry group.

For the four-bead, two-color bracelet with rotations only ($|G| = 4$): rotating by 0° fixes all 16 colorings; rotating by 90° or 270° fixes only the 2 colorings where all beads are the same color; rotating by 180° fixes the 4 colorings where opposite beads match. The average is $(16 + 2 + 4 + 2)/4 = 24/4 = 6$ — there are 6 genuinely distinct bracelets, not 16 and not 4.