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Let P be n-sided regular polygon. How many contained polygons can be constructed from any subset of P's n vertices such that no pair of polygons are congruent.

I've struggled with this problem for some time and unfortunately failed to find solution nor guidance on how to approach this problem. My first idea was to divide calculate the result as sum of k-element subsets where $$k \leq n$$
This approach did not lead me to a general formula, though is probably a start.

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    Let me clarify. Let $\mathcal P$ be the set of vertices of the $n$-gon. The question is about the maximal $N$ such that there are $N$ pairwise non-congruent polygons, whose vertices are in $\mathcal P$; these polygons may have common vertices. Correct? (In this formulation this is an application of Burnside's lemma.) – 2017-01-27
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    Yes, you could formulate this problem in such way. Well, then. I'm off to learn Burnside's lemma! – 2017-01-27
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    Search math.SE for "burnside polygon" or just "burnside lemma". There are some very similar questions with nice answers. Feel free to post your solution if you succeed in counting. – 2017-01-27

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