Table of

Furthermore, every point set in general position also has a minimum pseudotriangulation whose maximum face degree is four (i.e. each face of this pseudotriangulation has at most four vertices).
The counting results are summarized in the following table. For the point set(s) with the maximal number of minimum pseudotriangulations we give the number in the data base (table column order type # for max starting with 0 for the first set, which is in line with my tools provided below, but in contrast with Oswin Aichholzer's counting that starts with 1). The numbers are linked to the point coordinates. An example picture is available too. The last column gives access to the full binary data following the format of the order type data base.
We further observe in the data sets that the number of minimum pseudotriangulations is always greater than the number of triangulations except if the points are in convex position where the numbers are identical. Furthermore, the points in convex position minimize the number of minimum pseudotriangulations, which has actually now been proven to be correct for all number of points by O. Aichholzer, F. Aurenhammer, H. Krasser, and B. Speckmann in CCCG 2002, see here.
#points  #order types  min #pseudotr.  max #pseudotr.  order type # for max  data files 
3  1  1  1  set 0 (picture)  pseudo03.b08 
4  2  2  3  set 1 (picture)  pseudo04.b08 
5  3  5  13  set 2 (picture)  pseudo05.b08 
6  16  14  76  set 14 (picture)  pseudo06.b08 
7  135  42  485  set 124 (picture)  pseudo07.b16 
8  3315  132  3555  set 2990 (picture) set 3198 (picture)  pseudo08.b16 
9  158817  429  27874  set 151720 (picture)  pseudo09.b16 
10  14309547  1430  234160  set 13413893 (picture) set 13812359 (picture)  pseudo10.b32.gz 
Lutz Kettner, David Kirkpatrick, and Bettina Speckmann. Tight Degree Bounds for Pseudotriangulations of Points. In Proc. 13th Canad. Conf. on Computational Geometry, pp. 117120, 2001. (Abstract)
Hervé Brönnimann, Lutz Kettner, Michel Pocchiola, and Jack Snoeyink. Counting and Enumerating Pseudotriangulations with the Greedy Flip Algorithm. In preparation. 2001. (Abstract)
Lutz Kettner. PseudoTriangulation Workbench (ptw): User Manual. September 2001.
The PseudoTriangulation Workbench (ptw) is a research tool to investigate pseudotriangulations of points in the plane. It allows to create point sets, compute pseudotriangulations using various algorithms, interact with them using flips, highlight interesting properties using different color schemes, and enumerate all pseudotriangulations for a given point set. The enumeration can also be used with a branch&bound version to test hypothesis, such as minimal vertex degree. Currently, the algorithms and enumeration for pseudotriangulations create only minimal pseudotriangulations. Some support exists for normal triangulations, such as creation and diagonal flips. Commandline tools allow the enumeration of points sets, for example taken from the data base of order types for small point sets, without graphical user interface.