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A325002
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Triangle read by rows: T(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.
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8
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1, 2, 1, 2, 2, 1, 3, 3, 2, 1, 4, 6, 4, 2, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 2, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 2, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 2, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 2
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OFFSET
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1,2
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COMMENTS
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For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two. The only chiral pair occurs when k=n+1; for k <= n all the colorings are achiral.
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LINKS
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FORMULA
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T(n,k) = binomial(n,k-1) + [k==n+1] = A007318(n,k-1) + [k==n+1].
G.f. for row n: x * (1+x)^n + x^(n+1).
G.f. for column k>1: x^(k-1)/(1-x)^k + x^(k-1).
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EXAMPLE
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Triangle begins with T(1,1):
1 2
1 2 2
1 3 3 2
1 4 6 4 2
1 5 10 10 5 2
1 6 15 20 15 6 2
1 7 21 35 35 21 7 2
1 8 28 56 70 56 28 8 2
1 9 36 84 126 126 84 36 9 2
1 10 45 120 210 252 210 120 45 10 2
1 11 55 165 330 462 462 330 165 55 11 2
1 12 66 220 495 792 924 792 495 220 66 12 2
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 2
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 2
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 2
For T(3,2)=3, the tetrahedron may have one, two, or three faces of one color.
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MATHEMATICA
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Table[Binomial[n, k-1] + Boole[k==n+1], {n, 1, 15}, {k, 1, n+1}] \\ Flatten
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CROSSREFS
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Other n-dimensional polytopes: A325008 (orthotope), A325016 (orthoplex).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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