| The number of triangulations of the cyclic polytope C(n,n-4) (2007) | |||||||||||||||
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| We show that the exact number of triangulations of the cyclic polytope C(n; n \Gamma 4) is (n + 4)2 n\Gamma4 2 \Gamma n if n is even and i 3n+11 2 p 2 j 2 n\Gamma4 2 \Gamma 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 triangulations which use a specific simplex. We also compute a tight upper bound for the number of regular triangulations of C(n; n \Gamma 4) in terms of n. Introduction By a triangulation of a finite point set A ae R d we mean a simplicial complex geometrically realized in R d with vertex set contained in A and which covers the convex hull of A. If A is the vertex set of a polytope P this definition agrees with the standard definition of triangulation of P . The collection of all triangulations of a fixed point set has attracted attention in recent years for its connections to algebraic ... | |||||||||||||||
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