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A325000
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Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.
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13
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1, 3, 1, 6, 4, 1, 10, 10, 5, 1, 15, 20, 15, 6, 1, 21, 35, 35, 21, 7, 1, 28, 56, 70, 56, 28, 8, 1, 36, 84, 126, 126, 84, 36, 9, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1
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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 triangular 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 unoriented colorings are the same if congruent; chiral pairs are counted as one.
Note that antidiagonals are part of rows of the Pascal triangle.
T(n,k-n) is the number of chiral pairs of colorings of the facets (or vertices) of a regular n-dimensional simplex using k or fewer colors. - Robert A. Russell, Sep 28 2020
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LINKS
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FORMULA
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T(n,k) = binomial(n+k,n+1) = A007318(n+k,n+1).
T(n,k) = Sum_{j=1..n+1} A007318(n,j-1) * binomial(k,j).
G.f. for row n: x / (1-x)^(n+2).
Linear recurrence for row n: T(n,k) = Sum_{j=1..n+2} -binomial(j-n-3,j) * T(n,k-j).
G.f. for column k: (1 - (1-x)^k) / (x * (1-x)^k) - k.
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EXAMPLE
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The array begins with T(1,1):
1 3 6 10 15 21 28 36 45 55 66 78 91 105 ...
1 4 10 20 35 56 84 120 165 220 286 364 455 560 ...
1 5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 ...
1 6 21 56 126 252 462 792 1287 2002 3003 4368 6188 8568 ...
1 7 28 84 210 462 924 1716 3003 5005 8008 12376 18564 27132 ...
1 8 36 120 330 792 1716 3432 6435 11440 19448 31824 50388 77520 ...
1 9 45 165 495 1287 3003 6435 12870 24310 43758 75582 125970 203490 ...
1 10 55 220 715 2002 5005 11440 24310 48620 92378 167960 293930 497420 ...
...
For T(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors. For T(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
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MATHEMATICA
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Table[Binomial[d+1, n+1], {d, 1, 15}, {n, 1, d}] // Flatten
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CROSSREFS
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Unoriented: A007318(n,k-1) (exactly k colors), A327084 (edges, ridges), A337884 (faces, peaks), A325005 (orthotope facets, orthoplex vertices), A325013 (orthoplex facets, orthotope vertices).
Chiral: A327085 (edges, ridges), A337885 (faces, peaks), A325006 (orthotope facets, orthoplex vertices), A325014 (orthoplex facets, orthotope vertices).
Cf. A104712 (same sequence for a triangle; same sequence apart from offset).
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KEYWORD
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AUTHOR
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STATUS
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approved
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