The Number of Triangulations of the Cyclic Polytope C (n, n-4)

We show that the exact number of triangulations of the cyclic poly-tope C (n; n ? 4) is (n + 4)2 n?4 2 ? n if n is even and 3n+11 2 p 2 2 n?4 2 ? n if n is odd. These formulas were previously conjectured by the second author. Our techniques are based on Gale duality and the concept of virtual chambers. They further provide formulas for the number of triangu-lations which use a speciic simplex. We also compute a tight upper bound for the number of regular triangulations of C (n; n ? 4) in terms of n.