On numbers of pseudo-triangulations

We study the maximum numbers of pseudo-triangulations and pointed pseudo-triangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation.We derive the bounds O(5.45N)O(5.45N) and Ω(2.41N)Ω(2.41N) for the maximum number of pointed pseudo-triangulations that can be contained in a specific triangulation over a set of N   points. For the number of all pseudo-triangulations contained in a triangulation we derive the bounds O⁎(6.54N)O⁎(6.54N) and Ω(3.30N)Ω(3.30N). We also prove that O⁎(89.1N)O⁎(89.1N) pointed pseudo-triangulations can be embedded over any specific set of N   points in the plane, and at most 120N120N general pseudo-triangulations.