A355502 Inequivalent simultaneous colorings of the faces, vertices and edges of the cube under rotational symmetry using exactly n colors.

1, 2802750, 105904482864, 187226450755016, 61150982606571900, 6737855626357107000, 342689297671355738880, 9659365383584921484480, 169366933728740293383600, 1995772772375467764487200

Offset

1,2

Comments

This sequence is finite as we have slots for at most 26 colors.

Links

Table of n, a(n) for n=1..10.

Marko Riedel et al., Mathematics Stack Exchange, Coloring faces, vertices, edges of a cube.

Formula

a(n) = Sum_{q=0..n} C(n,q) (-1)^(n-q) A356685(q).

a(n) = (n!/24) * (S2(26,n) + 9*S2(14,n) + 8*S2(10,n) + 6*S2(8,n)) where S2 is the Stirling number of the second kind (Stirling set number).

Cycle index is (1/24) * (x1^26 + 6*x1^2*x4^6 + 9*x1^2*x2^12 + 8*x1^2*x3^8).

Crossrefs

Cf. A356685.

Sequence in context: A316487 A250535 A203912 * A356685 A234189 A321056

Adjacent sequences: A355499 A355500 A355501 * A355503 A355504 A355505

Keyword

nonn,fini

Author

Marko Riedel, Aug 22 2022

Status

approved