Counting n-gons

[franktaw at netscape.net]

I agree; the order of the sides should really be significant, up to 
rotation and reflection.

The sequence with this definition for quadrilaterals is A05788.  
Sequences for polygons with more sides do not seem to be in the OEIS.

---
I'm getting more interested in counting topologically distinct 
polygons, not necessarily simple.  To be definite, I'll allow crossings 
only in mid-edge, not at vertices, and no more than 2 edges can meet at 
a point.  Edges are identified by, and only by, their length.

For quadrilaterals, n = 4, the sides are 1,1,1,1.  There is no way to 
make these cross, so a(4) = 1.

For n = 5, the sides are 1,1,1,2.  The length 2 side can cross the side 
opposite, so a(5) = 2.

For n = 6, the sides are 1,1,2,2 and 1,2,1,2.  I'm having some trouble 
visualizing it, but I'm pretty sure the 1,1,2,2 does not admit of any 
crossings.  The 1,2,1,2 can have the length 2 sides cross, so a(6) = 3.

For n = 7, 1,1,2,3; 1,2,1,3; and 1,2,2,2 each admit a crossing: a(7) = 
6.

So the sequence starts (from n = 0) 0,0,0,0,1,2,3,6.  Not enough to 
look it up.

Is it true that any quadrilateral not of the form a,a,b,b admits a 
crossing?  If so, it should be possible to compute this sequence.

For pentagons, there are more possible crossing patterns (how many? - 
at least 5), making it more difficult.

Is the sequence of the number of possible crossing patterns for an 
n-gon in the OEIS?

Franklin T. Adams-Watters


-----Original Message-----
From: jvospost3 at gmail.com
...
The partitions can be the same, but the order of edges encountered in a 
circuit can differ.
...

On 10/24/06, T. D. Noe <noe at sspectra.com> wrote:

I just submitted A124278, triangle of the number of k-gons having 
perimeter
n.  
...
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