Geodesic
Counting \(d\)-polytopes with \(d+3\) vertices
The electronic journal of combinatorics, Tome 13 (2006)
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We completely solve the problem of enumerating combinatorially inequivalent $d$-dimensional polytopes with $d+3$ vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum's book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on similar preliminaries but using then a different enumeration method involving automata. In addition, we introduce and solve the problem of counting oriented and achiral (i.e., stable under reflection) $d$-polytopes with $d+3$ vertices. The complexity of computing tables of coefficients of a given size is then analyzed. Finally, we derive precise asymptotic formulas for the numbers of $d$-polytopes, oriented $d$-polytopes and achiral $d$-polytopes with $d+3$ vertices. This refines a first asymptotic estimate given by Perles.
DOI : 10.37236/1049
Classification : 52B11, 52B35, 05A15, 05A16
Mots-clés : polytope, combinatorial structure, achiral polytope 
@article{10_37236_1049,
     author = {\'Eric Fusy},
     title = {Counting \(d\)-polytopes with \(d+3\) vertices},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1049},
     zbl = {1093.52005},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1049/}
}
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Éric Fusy. Counting \(d\)-polytopes with \(d+3\) vertices. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1049

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