A140986
Number of n-colorings of the cubical graph.
Original entry on oeis.org
0, 0, 2, 114, 2652, 29660, 198030, 932862, 3440024, 10599192, 28478970, 68716010, 152040372, 313269684, 608134982, 1122341430, 1983307440, 3375066032, 5556852594, 8885943522, 13845350540, 21077015820, 31421193342, 45962742254, 66085098312, 93532729800
Offset: 0
- Eric M. Schmidt, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Chromatic Polynomial
- Eric Weisstein's World of Mathematics, Cubical Graph
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
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a:= n-> n^8 -12*n^7 +66*n^6 -214*n^5 +441*n^4 -572*n^3 +423*n^2 -133*n:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 01 2009
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A140986(n):=n^8-12*n^7+66*n^6-214*n^5+441*n^4-572*n^3 +423*n^2-133*n$
makelist(A140986(n),n,0,30); /* Martin Ettl, Nov 03 2012 */
A334159
Irregular triangle read by rows: T(n,k) is the number of colorings of the n-hypercube graph using exactly k unlabeled colors, k = 1..2^n.
Original entry on oeis.org
1, 0, 1, 0, 1, 2, 1, 0, 1, 18, 92, 146, 80, 16, 1, 0, 1, 494, 54583, 1507094, 12630906, 40096740, 58031885, 43419502, 18212138, 4498756, 670366, 60220, 3156, 88, 1, 0, 1, 197546, 5427041958, 17973998149410, 10961517110194516, 1450479305675145412, 56507865332978414188
Offset: 0
Triangle begins:
0 | 1;
1 | 0, 1;
2 | 0, 1, 2, 1;
3 | 0, 1, 18, 92, 146, 80, 16, 1;
4 | 0, 1, 494, 54583, 1507094, 12630906, 40096740, 58031885, 43419502, 18212138, 4498756, 670366, 60220, 3156, 88, 1;
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
A334278
Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the cubical graph Q_n, 0 <= k <= 2^n.
Original entry on oeis.org
0, 1, 0, -1, 1, 0, -3, 6, -4, 1, 0, -133, 423, -572, 441, -214, 66, -12, 1, 0, -3040575, 14412776, -31680240, 43389646, -41821924, 30276984, -17100952, 7701952, -2794896, 818036, -191600, 35264, -4936, 496, -32, 1
Offset: 0
Table begins:
n/k| 0 1 2 3 4 5 6 7 8
---+-------------------------------------------
0| 0, 1
1| 0, -1, 1
2| 0, -3, 6, -4, 1
3| 0, -133, 423, -572, 441, -214, 66, -12, 1
Cf.
A296914 is the reverse of row 3.
Cf.
A334279 is analogous for the n-dimensional cross-polytope, the dual of the n-cube.
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with(GraphTheory): with(SpecialGraphs):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
ChromaticPolynomial(HypercubeGraph(n), x)):
seq(T(n), n=0..4); # Alois P. Heinz, Jan 14 2025
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T[n_, k_] := Coefficient[ChromaticPolynomial[HypercubeGraph[n], x], x, k]
Showing 1-4 of 4 results.
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