On the number of drawings of a combinatorial triangulation

In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and n additional vertices. In this note, we study how many ways a combinatorial triangulation can be drawn as geometric triangulation, that is, with straight-line segments, on a given point set in the plane. Our central contribution is that there exists a combinatorial triangulation with n vertices that can be drawn in at least Ω(1,31ⁿ) ways on a set of n points as different geometric triangulations. We also show an upper bound on the number of drawings of a combinatorial triangulation on the so-called double chain point set.