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.