Counting polygon triangulations is hard.
D. Eppstein.
arXiv:1903.04737.
Proc. 35th Int. Symp. on Computational Geometry, Portland, Oregon, June 2019.
Leibniz International
Proceedings in Informatics (LIPIcs) 129, 2019, pp. 33:1–33:17, doi:10.4230/LIPIcs.SoCG.2019.33.
Discrete Comput. Geom. 64 (4): 1210–1234, 2020 (special issue for SoCG 2019), doi:10.1007/s00454-020-00251-7.
Given a polygon with holes, it is \(\#\mathsf{P}\)-complete to determine how many triangulations it has.
(UCLA seminar talk slides – SoCG slides – Blog post: Counting polygon triangulations is hard)