4
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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):

$n=d+1$

A) 1 B) 1 C) 1

$n=d+2$

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

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

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

$n=d+3$

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

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.