# necklace reconstruction in the permutation case

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.

• Are the subsets allowed to intersect? Or interleave? – მამუკა ჯიბლაძე Sep 26 '17 at 17:12
• Does cyclic order come with direction? E.g., would cyclic orders 132 and 231 be considered the same? – Max Alekseyev Sep 27 '17 at 0:08
• Yes, 132 and 231 are different (hence "necklace", but these things are not as standard as they might be, so I will clarify in the question). – Karen Yeats Sep 27 '17 at 18:32
• In the application I have in mind the subsets would typically intersect and could potentially interleave. – Karen Yeats Sep 27 '17 at 18:36