Counting n-gons

[Jonathan Post]

I agree with Franklin, who is asking the right questions in the right way.
Of course, if one is slipping towards topology, there is also the question
of whether one is embedding the polygons in the Euclidean plane R^2, the
3-space R^3, or some other manifold such as on a Moebius strip, projective
plane, Klein bottle....
I have worked on enumerating polyheptagons in the Klein quartic
(topologically a 3-holed torus) as I commented in
A119611<http://www.research.att.com/%7Enjas/sequences/A119611>
 Number of free polyominoes in {4,5} tesselation of the hyperbolic plane.
This matter of embedding in R^3 also relates to
 A122059 <http://www.research.att.com/%7Enjas/sequences/A122059>
 Number of different polygonal *knots* with n straight line segments.
-- Jonathan Vos Post

On 10/24/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> 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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