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

Author

Frank M Jackson, Aug 12 2018

Keywords

• nonn
• fini

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.

a(6) = 17577...66368*10^146 has 1405 digits. - Georg Fischer, Jun 16 2025

Examples

			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.
		

Links

• Georg Fischer, Table of n, a(n) for n = 1..5
• Wikipedia, Regular 4-polytope

Crossrefs

Cf. A063924, A098427, A198861, A273509.

Programs

Maple

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

Mathematica

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

Formula

a(n) = 2*A063924(n)! / A273509(n). [Corrected by Georg Fischer, Jun 16 2025]