A000543 -id:A000543 - cp's OEIS frontend

A334616 Number of 2-colorings of an n X n X n grid, up to rotational symmetry.

2, 23, 5605504, 768614338020786176, 1772303994379887844373479205703254016, 4388012152856549445746584486819723041078276071004502223505850368, 746581580725934736852480760189481426040548499078234470565449222456544381939194522144498021170453413888

Offset: 1

Paul Oelkers, Sep 08 2020

nonn

The cycle index of the permutation group is given by:
Even n: (1/24)*(s_1^n^3 + 8*s_1^n*s_3^((n^3-n)/3) + 6*s_2^(n^3/2) + 6*s_4^(n^3/4) + 3*s_2^(n^3/2));
Odd n: (1/24)*(s_1^n^3 + 8*s_1^n*s_3^((n^3-n)/3) + 6*s_1^n*s_2^((n^3-n)/2) + 6*s_1^n*s_4^((n^3-n)/4) + 3*s_1^n*s_2^((n^3-n)/2)).

			a(2)=23 from:
  00 00
  00 00
------------------------------------------
  10 00
  00 00
------------------------------------------
  11 00   10 00   10 01   10 00
  00 00   01 00   00 00   00 01
------------------------------------------
  11 00   11 00   01 10
  10 00   00 10   10 00
------------------------------------------
  11 00   11 00   01 10   11 00   11 10
  11 00   10 01   10 01   00 11   10 00
------------------------------------------
  00 11   00 11   10 01
  01 11   11 01   01 11
------------------------------------------
  00 11   01 11   01 10   01 11
  11 11   10 11   11 11   11 10
------------------------------------------
  01 11
  11 11
------------------------------------------
  11 11
  11 11
------------------------------------------
An example for the 2-coloring of the 3 X 3 X 3 grid can be written as:
  110 000 111
  100 000 111
  100 000 111
This coloring is equivalent to:
  111 000 111
  001 000 111
  000 000 111
  because you can get this configuration by rotating the first coloring by 90 degrees.
But it is different from:
  011 000 111
  001 000 111
  001 000 111
  because reflections are not considered.

Wikipedia, Cycle index
Paul Oelkers, Hand written notes and sketches

This is the three-dimensional version of A047937.

Cf. A000543.

a(n) = (1/24)*(2^n^3 + 6*2^((n^3)/4) + 9*2^((n^3)/2) + 8*2^((n^3-n)/3+n)) for n even;

a(n) = (1/24)*(2^n^3 + 6*2^(((n^3)-n)/4+n) + 9*2^(((n^3)-n)/2+n) + 8*2^(((n^3-n)/3)+n)) for n odd.

More terms from Stefano Spezia, Sep 09 2020

A237748 Number of different ways to color the vertices of an n-dimensional hypercube using at most 2 colors.

Original entry on oeis.org

4, 6, 23, 496, 2275974, 800648638402240, 1054942853799126580390222487977120, 22436153203535039105819651040959324360753617244078062654624560815030272

Offset: 1

Seiichi Azuma, Feb 12 2014

nonn

Two colorings are regarded as the same if they are conjugated by a permutation of the vertices caused by a rotation of the hypercube.

Regarding mirrored coloring also as the same, the number of ways appears to be given by A000616. - Seiichi Azuma, Feb 14 2014

			For n=2, there are 6 patterns to color the vertices of the square:
..1..1....1..1....1..1....1..0....1..0....0..0
..1..1....1..0....0..0....0..1....0..0....0..0
For example, the next pattern is regarded as the same with the 3rd one above:
..1..0
..1..0

Seiichi Azuma, Number of different ways to color vertices of a n-dimensional hypercube using at most K colors

Cf. A000543.

Column 2 of A325012.

A316093 Non-isomorphic colorings of the cube under rotations, using at most N colors on the faces and M colors on the vertices. Square array H(N,M) with N,M > 0 read by antidiagonals.

Original entry on oeis.org

1, 10, 23, 57, 776, 333, 240, 8121, 17946, 2916, 800, 44608, 200961, 176160, 16725, 2226, 168675, 1124208, 1995852, 1045050, 70911, 5390, 501528, 4281300, 11198720, 11877825, 4485960, 241913, 11712, 1261701, 12773538, 42697300, 66700400, 51044337, 15385706, 701968, 23355, 2807296, 32195646, 127461216, 254387500, 286724160, 175153881, 44761216, 1798281, 43450, 5685903, 71718336, 321364540, 759518850, 1093653675, 983988208, 509689776, 114826410, 4173775

Offset: 1

Marko Riedel, Jun 24 2018

			Square array begins:
     1,     10,      57,      240,      800, ...
    23,    776,    8121,    44608,   168675, ...
   333,  17946,  200961,  1124208,  4281300, ...
  2916, 176160, 1995852, 11198720, 42697300, ...

Marko Riedel et al., coloring cube sides and vertices

H(N,1) (first row) is A047780. H(1,M) (first column) is A000543.

H(N,M) = (1/24) (N^6 M^8 + 6 N^3 M^2 + 3 N^4 M^4 + 8 N^2 M^4 + 6 N^3 M^4).

Cycle index is (1/24)*(a1^6 b1^8 + 6 a1^2 a4 b4^2 + 3 a1^2 a2^2 b2^4 + 8 a3^2 b1^2 b3^2 + 6 a2^3 b2^4).

cp's OEIS Frontend

A334616 Number of 2-colorings of an n X n X n grid, up to rotational symmetry.

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A237748 Number of different ways to color the vertices of an n-dimensional hypercube using at most 2 colors.

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A316093 Non-isomorphic colorings of the cube under rotations, using at most N colors on the faces and M colors on the vertices. Square array H(N,M) with N,M > 0 read by antidiagonals.

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