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Number of n-dimensional polytopes with vertices from {0,1}^n up to (0,1)-equivalence.
3

%I #15 Jun 21 2023 01:22:08

%S 1,2,12,347,1226525,400507800465455

%N Number of n-dimensional polytopes with vertices from {0,1}^n up to (0,1)-equivalence.

%C a(6) = 400507800465455 is obtained as the sum of the number of 6-dimensional polytopes with up to 12 vertices, 94826705, given in the footnote in p. 119 of Aichholzer (2000), and the counts for k > 12 vertices F_6(k) given in tables 3, 6 and 7 of Chen & Guo (2014). Chen & Guo have typos: F_5(29) in table 2 should be 10 and F_5(16) in table 5 should be 169110 (cf. polyDB or Aichholzer's table 2). - _Andrey Zabolotskiy_, Jun 20 2023

%H Oswin Aichholzer, <a href="https://doi.org/10.1007/978-3-0348-8438-9_5">Extremal Properties of 0/1-Polytopes of Dimension 5</a>. In: Polytopes - Combinatorics and Computation, DMV Seminar vol 29, Birkhäuser, Basel, 2000.

%H William Y. C. Chen and Peter L. Guo, <a href="https://doi.org/10.1007/s00454-014-9630-5">Equivalence Classes of Full-Dimensional 0/1-Polytopes with Many Vertices</a>, Discrete Comput. Geom., 52 (2014), 630-662.

%H Andreas Paffenholz and the polymake team, <a href="https://polydb.org">polyDB</a>. See Polytopes > Geometric Polytopes > 0/1 Polytopes by Oswin Aichholzer.

%H Chuanming Zong, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01050-5">What is known about unit cubes</a>, Bull. Amer. Math. Soc., 42 (2005), 181-211.

%Y Cf. A105230, A105232.

%K nonn,hard,more

%O 1,2

%A _N. J. A. Sloane_, Apr 16 2005

%E a(6) from _Andrey Zabolotskiy_, Jun 20 2023