P. Lisonek,
Closed forms for the number of polygon dissections.
Journal of Symbolic Computation 20 (1995), 595-601.
We consider unlabeled dissections of the regular $s$-gon
into $r$ cells by means of non-intersecting diagonals.
We prove that if the parameter $r$ is fixed
then the number of dissections is quasi-polynomial in $s$.