A334247
Number of acyclic orientations of the edges of an n-dimensional cube.
Original entry on oeis.org
1, 2, 14, 1862, 193270310, 47171704165698393638
Offset: 0
For n=2, there are 14 ways to orient the edges of a square without cycles (see links).
Cf.
A334248 is the number of acyclic orientations with rotations and reflections of the same orientation excluded.
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with(GraphTheory): with(SpecialGraphs):
a:= n-> abs(ChromaticPolynomial(HypercubeGraph(n), -1)):
seq(a(n), n=0..4); # Alois P. Heinz, Jan 14 2025
A334358
Irregular triangle read by rows: row n gives scaled coefficients of the chromatic polynomial corresponding to colorings of the n-hypercube graph up to automorphism, highest powers first, 0 <= k <= 2^n.
Original entry on oeis.org
1, 0, 1, -1, 0, 1, -2, 3, -2, 0, 1, -12, 72, -256, 579, -812, 644, -216, 0, 1, -32, 496, -4936, 35276, -191840, 820328, -2808636, 7759343, -17276144, 30675244, -42494732, 44214736, -32375904, 14772272, -3125472, 0, 1, -80, 3160, -82080, 1575420, -23805776, 294640000
Offset: 0
Triangle begins:
0 | 1, 0;
1 | 1, -1, 0;
2 | 1, -2, 3, -2, 0;
3 | 1, -12, 72, -256, 579, -812, 644, -216, 0;
...
The corresponding polynomials are:
x;
(x^2 - x)/(1!*2^1);
(x^4 - 2*x^3 + 3*x^2 - 2*x)/(2!*2^2);
(x^8 - 12*x^7 + 72*x^6 - 256*x^5 + 579*x^4 - 812*x^3 + 644*x^2 - 216*x)/(3!*2^3);
...
The polynomial (x^4 - 2*x^3 + 3*x^2 - 2*x)/(2!*2^2) gives A002817 when evaluated at integer values of x.
A334304
Number of distinct acyclic orientations of the edges of an n-dimensional cube with complete graphs as facets.
Original entry on oeis.org
For n=2, the n-dimensional cube is a square, and the (n-1)-dimensional facets are the edges of the square. Replacing the edges with complete graphs on 2 vertices does not change the graph.
There are 3 distinct (under rotations and reflections) acyclic orientations of the edges of this graph:
*->-* *->-* *-<-*
| | | | | |
^ ^ ^ v ^ v
| | | | | |
*->-* *->-* *->-*
Therefore a(2) = 3.
For n=3, the n-dimensional cube is a cube, and the (n-1)-dimensional facets are the faces of the cube. Replacing the faces with complete graphs on 4 vertices gives the graph that is the edges of a cube with diagonal edges added to each face (the "16-cell"). a(3) is the number of distinct acyclic orientations of this graph.
A334248 is the number of distinct acyclic orientations of a n-cube (without the addition of complete graphs).
A000012 is the number of reference elements needed when using the finite element method for an n-dimensional problem with simplectic cells.
Showing 1-3 of 3 results.
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