When objects can be rotated or reflected into one another, naive counting overcounts by the size of the symmetry group acting on the configuration — unless that group acts freely (no nontrivial symmetry fixes any arrangement), in which case simply divide by ∣G∣.
For linear-to-circular arrangements of k distinct items, fixing one item kills rotational symmetry: there are (k−1)! circular orders.
When the group does not act freely (e.g. colourings, where some rotations fix many colourings), use Burnside’s lemma:
#{orbits}=∣G∣1g∈G∑∣Fix(g)∣,
summing, for each symmetry g, the number of colourings unchanged by g. For rotations of a cycle of length n by d steps, ∣Fix(g)∣=cgcd(n,d) for c colours; for bracelets also include the n reflections. For labelling objects with all-distinct labels and a symmetry group of order ∣G∣ acting freely, the count is simply n!/∣G∣.