I'm new to mathoverflow but hopefully anyone here can point me in the right direction.


The problem is as follows, imagine you have 4 beads, lets give them numbers 1,2,3,4. Now I want the unique `necklaces` without rotation, mirror nor duplication of the beads. So 1234 is the same as 3412 and 4321.


How do you call this sequence mathematically? It is related to a `free necklace` and to `bracelet`, but those include duplicate use of beads.



```
For N=1: 1
For N=2: 12 (same as 21)
For N=3: 123 (same as 132,231,213,321,312)
For N=4: 1234 1243 1324 (so not 1342 1423 1432, those are equal flipped)

```

Is there a good mathematical term for this set? And second: Is there a good way to construct the set for a given number?



Terminology for beads/necklace/bracelet problem
I don't know the name for this set, but it seems straightforward to generate it. In each equivalence class there is a unique representative starting with "1", and such that either the position of "2" is less than $\frac{N+2}{2}$, or it is equal to $\frac{N+2}{2}$ and the position of "3" is less than $\frac{N+2}{2}$. You just have to generate all the permutations of $(1,\dots,N)$ subject to these conditions.


