@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?
@johncarlosbaez
actually I guess for the purposes of the "divide square into similar rectangles" problem we could just allow the side length of the interior rectangle to be negative
