The number of triangulations of a convex $n$-gon is $C\_{n-2}$ the $n-2$nd Catalan number. What I am wondering, is if there is a way to enumerate the isomorphism types of these as graphs? I am currently working an a project where I want to only consider all possible unlabeled triangulations of $n$-gons as a way to classify an $n$-gon. Clearly $C\_{n-2}$ is an upper bound, but I haven't been able to make or find any progress on that thus far.



Number of isomorphism classes of triangulations of a convex polygon
If I understand correctly, these are [A001683](http://oeis.org/A001683) if turning over the $n$-gon is not allowed as an isomorphism, and [A000207](http://oeis.org/A000207) if it is. In both articles there are formulae.


You might also be interested that [plantri](http://cs.anu.edu.au/~bdm/plantri/) can compute these isomorphism classes very quickly, several hundred thousand per second. The argument lists for triangulating an $n$-gon are "plantri -P# -c2m2 -o #" and "plantri -P# -c2m2 #" respectively, where # is the value of $n$.


ADDED: Every abstract isomorphism between two triangulations preserves the $n$-gon, for the following reason. The $n$-gon is a hamiltonian cycle in the graph. Every chord is a separating edge (removing its two vertices disconnects the graph) and such edges cannot lie on a hamiltonian cycle. I.e., the $n$-gon is the unique hamiltonian cycle.


