@johncarlosbaez
oh i just accidentally ran into these the number of rooted 2 connected planar graphs with n edges (allowing multiple edges between vertices) runs as G_2=1,2,6,22,91 whereas the schroder numbers run 1,2,6,22,91.

roughly you can build 2-connected planar graphs out of 3-connected planar graphs and the schroder numbers count graphs which are "trivial" in this sense

@johncarlosbaez
(oops schoder numbers run 1,2,6,22,90; typo)

@johncarlosbaez
the first nontrivial graph is the tetrahedron which explains the 90-91 difference

@alexthecamel - ah, the tetrahedron! Nice.

@johncarlosbaez
roughly the idea is to treat vertical lines as vertices and boxes as edges. thing is, this shape you had before gives the tetrahedron with this rule, but there are 2 such shapes.
so to do this in more generality we'd need to think of these as directed graphs and add some more criteria?

@alexthecamel - Cool thought. But the center is square so the outside ones must be too. This forces the outside ones to have side length 1/2 that of the whole square. This forces the central one to have side length 0.

@johncarlosbaez
in this particular case the side length must be 0
but for more general configurations which you can't get from guillotine cuts it seems like a useful idea

@johncarlosbaez
(I think "some side length is zero" is the only way things can break if we take x= a +ve real root of the polynomial we get, side lengths being negative should still get you a well behaved dissection)

@alexthecamel - interestingly, side lengths being negative still gives a positive area. Can you give me an example of a dissection that involves negative edge lengths? Do the areas add up to the area of the square or rectangle being dissected?