Abstract
We show that the exact number of triangulations of the standard cyclic polytope C(n,n-4) is (n+4)2 (n-4)/2 -n if n is even and \left((3n+11)/2\right)2 (n-5)/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 chamber. They further provide formulas for the number of triangulations which use a specific simplex. We also compute the maximum number of regular triangulations among all the realizations of the oriented matroid of C(n,n-4) .
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