# Counting rectilinear polygons

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.

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squeeze mapping? – Will Jagy Dec 6 '11 at 0:28
The number of vertical edges doesn't necessarily equal the number of horizontal edges unless you make some assumption about the shape of the polygon. (E.g.a polygon consisting of seven unit squares arranged in a C-shape.) – Hugh Thomas Dec 6 '11 at 1:06
@Will: Horizontal or vertical scaling, e.g., $x \mapsto \frac{1}{2} x$. – Joseph O'Rourke Dec 6 '11 at 1:07
@Hugh: I think you are counting edges differently? Any C-shape is composed of 4 horizontal and 4 vertical edges, where each edge is a segment perhaps of different length. – Joseph O'Rourke Dec 6 '11 at 1:11
Counting left right turn sequences might be a good start. Gerhard "Ask Me About System Design" Paseman, 2011.12.05 – Gerhard Paseman Dec 6 '11 at 1:44

Or to be more explicit, if we are interpreting the problem correctly, then the L-shapes consisting of an $n\times1$ rectangle upright with a $2\times1$ base all have 6 sides and are inequivalent for distinct $n$. – Gerry Myerson Dec 6 '11 at 5:05
If I'm reading the OP correctly, the condition was that there are $2n$ horizontal and $2n$ vertical segments, so the L-shapes are not valid examples. Fortunately, this isn't really an issue, as we can just use a larger staircase polygon. – Michael Biro Dec 6 '11 at 5:18