Suppose I have a regular n-gon. I want to draw some noncrossing diagonals to subdivide it into smaller polygons. In how many ways can I do this? The vertices are unlabeled, so I don't distinguish between rotations or reflections of a given subdivision.


A triangle has 1 subdivision (do nothing!); a square has 2, a pentagon has 3, and a hexagon has at least 9 -- I'm not certain that I haven't missed any.


In fact, what I would really like is not just a count, but an algorithm for generating such subdivisions. There are obvious algorithms that generate some subdivisions multiple times, but what I'd really like is an algorithm that only generates distinct subdivisions, and that generates all of them.



Number of subdivisions of an n-gon
I think this could be [A001004](http://www.research.att.com/~njas/sequences/A001004) in Sloane's Encyclopedia. It's hard to be sure without checking the references given there; the sequence is defined as "Number of symmetric dissections of a polygon.", which may or may not be what you mean. (In particular, the OEIS claims the next term is 20 and I'm afraid to try to check that.)


