Micha Perles used Gale diagrams to compute the number of simplicial $d$-polytopes with $d+3$ vertices and of general $d$-polytopes with $d+3$ vertices. The computation can be found in Chapter 6.3 of Grunbaum's book, "Convex polytopes". My question is for formulas for the number of simplicial $d$-polytopes with $d+3$ labelled vertices and general $d$-polytopes with $d+3$ labelled vertices. The Gale-diagram technique should apply (and probably be easier) but I am not aware of this being done.

One motivation would be in trying to compute the number of simplicial d-polytopes, triangulations of $(d-1)$-spheres (and related non simplicial objects) with $d+4$ labelled vertices.


With Ron Adin we worked out the answer for the simplicial case. A) All simplicial order types, B) simplicial polytopes, C) Neighborly polytopes ($d$ even):


A) 1 B) 1 C) 1


A) $2^{d+1}-1$,

B) $2^{d+1}-(d+3)$,

C) $\frac {1}{2} {{d+2}\choose {(d+2)/2}}$.


A) $C +1/2 \sum_{t=4}^n t! S(n-1,t)$.

B) $C'+ 1/2 \sum_{t=4}^n t! S(n-1,t)$.

Where $C=2 S(n-1,2) + 6 S(n-1,3)$ and $C' = 6S(n-1,3) - 4(n-1)S(n-2,2) + (n-1)(n-2)$, and $S$ is the Stirling number of the second kind.

C) $(n-1)!/2$.


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