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.



Counting rectilinear polygons
The number of right turns must always be 4 more than the number of left turns. This is the only constraint on turn sequences. Proof: Take the minimal impossible sequence satisfying this property. It must contain RR, so it is either RRRR or contains RRL. Delete the RL, find an example, and then add a small "tab" off the corner. If it's RRRR, draw a rectangle.


The number of words like that should be a fairly standard combinatorics problem. I don't know the exact formula.


Already with 6 turns, though, the number of distinct polygons is infinite. In assembling an L-shape, I have 6 degrees of freedom counting translation, and you only have a 4-dimensional space of symmetries.


