A317978 The number of ways to paint the cells of the six convex regular 4-polytopes using exactly n colors where n is the number of cells of each 4-polytope.

2, 210, 108972864000, 1077167364120207360000

Offset

1,1

Comments

Let G, the group of rotations in 4 dimensional space, act on the set of n! paintings of each convex regular 4-polytopes having n cells. There are n! fixed points in the action table since the only element in G that leaves a painting fixed is the identity element. The order of G is A273509/2. So by Burnside's Lemma a(n)=n!/|G|. a(5) = 120!/7200 and a(6) = 600!/72000 and they are too large to display.

See A198861 for the Platonic solids which are the analogs of the regular polyhedra in three dimensions.

Links

Table of n, a(n) for n=1..4.

Wikipedia, Regular 4-polytope

Formula

a(n) = 2*n!/A273509(n).

Example

The second of these six 4-polytopes (in sequence of cell count) is the 4-cube (with 8 cells). It has |G| = 192 rotations with n = 8. Hence a(2) = 8!/192 = 210.

Mathematica

{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}

Crossrefs

Cf. A098427, A198861, A273509.

Sequence in context: A092700 A178388 A056065 * A050445 A167833 A167838

Adjacent sequences: A317975 A317976 A317977 * A317979 A317980 A317981

Keyword

nonn,fini

Author

Frank M Jackson, Aug 12 2018

Status

approved