Suppose I want a necklace with $n$ beads labelled (bijectively) by $\{1, 2, \ldots n\}$, that is I want a cyclic order on $\{1, 2, \ldots, n\}$ (so for example $132$ is the same cyclic order as $321$ but different from $231$). Now suppose I know the cyclic order of some subsets of $\{1, 2, \ldots, n\}$ as they should appear in the necklace. I want to know what conditions on the subsets and their cyclic orders would make the necklace uniquely reconstructible, or given a set of subsets with orders how many compatible necklaces there are.

I feel like surely this is well studied (somewhere in the world of reconstructibility or elsewhere), but I'm not managing to find any results on it, so perhaps you can help me find where people have looked at this.

Thanks.