

A317978


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


1




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 4polytopes 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 4polytope


FORMULA

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


EXAMPLE

The second of these six 4polytopes (in sequence of cell count) is the 4cube (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



