Abstract 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 in 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 exist a combinatorial triangulation T and a point set S such that T can be drawn on S in at least (1, 31ⁿ) Ω (1, 31 n) ways 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.
Cruces et al. (Wed,) studied this question.