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I feel that this is a combinatorial question, I just don't know how to go about counting this.

The way I'm thinking about it is you have a list of $n$ numbers that are all 0 or 1. $\{0,1,1,0,0,0...,1\}$. Normally you would have $2^n$ choices but this is a little bit different since some orientations will be the same. The complement $\{1,0,0,1,1,1...,0\}$ is the same and any rotations (ie just shifting the list) of either one is also the same. Does anybody know a good way to approach counting this?

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    Please, give defenition of cycle orientation. – 2017-02-05
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    Are you asking about the number of non-isomorphic [orientations](https://en.wikipedia.org/wiki/Orientation_(graph_theory)) of the [cycle graph](https://en.wikipedia.org/wiki/Cycle_graph) $C_n$? – 2017-02-05
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    Yes, that is a good way to put it. Thanks! – 2017-02-05
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    There is a tag `necklace-and-bracelets` for [combinatorial problems of this kind](http://math.stackexchange.com/questions/tagged/necklace-and-bracelets). – 2017-02-05
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    Thank you for helping me clarify what I meant. The problem I was working on regards Hamiltonian cycles, so it made it harder for me to realize what I was really asking. – 2017-02-05

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