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.
Original entry on oeis.org
2, 210, 108972864000, 1077167364120207360000
Offset: 1
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.
-
{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200};
-
{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}
A309442
Minimum number of colors needed to color the cells of the six regular convex polychora such that no two cells with a common face share the same color (in the order 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, 600-cell).
Original entry on oeis.org
5, 4, 2, 3, 5, 3
Offset: 1
a(1) = 5, since in the 5-cell, each cell has a common face with every other cell (analogous to the tetrahedron, where each face has a common edge with every other face).
a(2) = 4, since in the 8-cell, each cell has a common face with every other cell except its "opposite" cell (analogous to the cube, where each face has a common edge with every other face except its opposite face).
a(3) = 2, since the 16-cell's dual graph has no odd-edge cycles (analogous to the octahedron's dual graph having no odd-edge cycles).
a(4) = 3, since the 24-cell has at least one 3-color solution, and its dual graph has a 3-vertex subgraph with no 2-color solution.
a(5) = 5, since the 120-cell has at least one 5-color solution, and its dual graph has a 30-vertex subgraph with no 4-color solution.
a(6) = 3, since the 600-cell has at least one 3-color solution, and its dual graph has a 5-vertex subgraph with no 2-color solution.
Showing 1-2 of 2 results.
Comments