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A325005
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Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.
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11
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1, 3, 1, 6, 6, 1, 10, 21, 10, 1, 15, 55, 56, 15, 1, 21, 120, 220, 126, 21, 1, 28, 231, 680, 715, 252, 28, 1, 36, 406, 1771, 3060, 2002, 462, 36, 1, 45, 666, 4060, 10626, 11628, 5005, 792, 45, 1, 55, 1035, 8436, 31465, 53130, 38760, 11440, 1287, 55, 1
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OFFSET
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1,2
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COMMENTS
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Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.
Also the number of unoriented colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.
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LINKS
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FORMULA
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A(n,k) = binomial(n + binomial(k+1,2) - 1, n).
A(n,k) = Sum_{j=1..2n} A325009(n,j) * binomial(k,j).
G.f. for row n: Sum_{j=1..2n} A325009(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).
G.f. for column k: 1/(1-x)^binomial(k+1,2) - 1.
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EXAMPLE
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Array begins with A(1,1):
1 3 6 10 15 21 28 36 45 55 ...
1 6 21 55 120 231 406 666 1035 1540 ...
1 10 56 220 680 1771 4060 8436 16215 29260 ...
1 15 126 715 3060 10626 31465 82251 194580 424270 ...
1 21 252 2002 11628 53130 201376 658008 1906884 5006386 ...
1 28 462 5005 38760 230230 1107568 4496388 15890700 50063860 ...
1 36 792 11440 116280 888030 5379616 26978328 115775100 436270780 ...
1 45 1287 24310 319770 3108105 23535820 145008513 752538150 3381098545 ...
For A(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses one color for each vertex.
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MATHEMATICA
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Table[Binomial[Binomial[d-n+2, 2]+n-1, n], {d, 1, 11}, {n, 1, d}] // Flatten
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CROSSREFS
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Other n-dimensional polytopes: A325000 (simplex), A325013 (orthoplex).
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KEYWORD
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AUTHOR
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STATUS
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approved
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