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A325007
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Array read by descending antidiagonals: A(n,k) is the number of achiral 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, 2, 1, 3, 6, 1, 4, 18, 10, 1, 5, 40, 55, 15, 1, 6, 75, 200, 126, 21, 1, 7, 126, 560, 700, 252, 28, 1, 8, 196, 1316, 2850, 1996, 462, 36, 1, 9, 288, 2730, 9261, 11376, 5004, 792, 45, 1, 10, 405, 5160, 25480, 50127, 38550, 11440, 1287, 55, 1, 11, 550, 9075, 61776, 181027, 225225, 116160, 24310, 2002, 66, 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. An achiral coloring is identical to its reflection.
Also the number of achiral 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(binomial(k+1,2) + n-1, n) - binomial(binomial(k,2),n).
A(n,k) = Sum_{j=1..2*n} A325011(n,j) * binomial(k,j).
G.f. for row n: Sum{j=1..2*n} A325011(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2*n} binomial(-2-j,2*n-j) * T(n,k-1-j).
G.f. for column k: 1/(1-x)^binomial(k+1,2) - (1+x)^binomial(k,2).
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EXAMPLE
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Array begins with A(1,1):
1 2 3 4 5 6 7 8 9 10 ...
1 6 18 40 75 126 196 288 405 550 ...
1 10 55 200 560 1316 2730 5160 9075 15070 ...
1 15 126 700 2850 9261 25480 61776 135675 275275 ...
1 21 252 1996 11376 50127 181027 559728 1529892 3784627 ...
1 28 462 5004 38550 225225 1053304 4119648 13942908 41918800 ...
1 36 792 11440 116160 881595 5263336 25794288 107427420 390891160 ...
For a(2,2)=6, all colorings are achiral: two with just one of the colors, two with one color on just one edge, one with opposite colors the same, and one with opposite colors different.
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MATHEMATICA
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Table[Binomial[Binomial[d-n+2, 2]+n-1, n]-Binomial[Binomial[d-n+1, 2], n], {d, 1, 11}, {n, 1, d}] // Flatten
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PROG
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(PARI) a(n, k) = binomial(binomial(k+1, 2)+n-1, n) - binomial(binomial(k, 2), n)
array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
/* Print initial 6 rows and 8 columns of array as follows: */
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CROSSREFS
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Other n-dimensional polytopes: A325001 (simplex), A325015 (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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