This is likely a very easy counting question inspired by some elementary geometry:

Consider a simple rectilinear polygon embedded in a plane in such a way that each of its edges is parallel to one of the coordinate axis. Two such polygons are considered *distinct* if they are not related by some composition of translation, scalar multiplication and squeeze mapping.

I would like to asses the number of such distinct simple rectilinear polygons which have $2n$ horizontal (equivalently vertical) edges for any chosen $n\in\mathbb N$.

Thank you.