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A324999
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Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.
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10
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1, 4, 1, 9, 4, 1, 16, 11, 5, 1, 25, 24, 15, 6, 1, 36, 45, 36, 21, 7, 1, 49, 76, 75, 56, 28, 8, 1, 64, 119, 141, 127, 84, 36, 9, 1, 81, 176, 245, 258, 210, 120, 45, 10, 1, 100, 249, 400, 483, 463, 330, 165, 55, 11, 1, 121, 340, 621, 848, 931, 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 oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
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LINKS
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FORMULA
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A(n,k) = binomial(n+k,n+1) + binomial(k,n+1).
A(n,k) = Sum_{j=1..n+1} A325002(n,j) * binomial(k,j).
G.f. for row n: (x + x^(n+1)) / (1-x)^(n+2).
Linear recurrence for row n: A(n,k) = Sum_{j=1..n+2} -binomial(j-n-3,j) * A(n,k-j).
G.f. for column k: (1 - 2*(1-x)^k + (1-x^2)^k) / (x*(1-x)^k) - 2*k.
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EXAMPLE
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The array begins with A(1,1):
1 4 9 16 25 36 49 64 81 100 121 144 169 196 ...
1 4 11 24 45 76 119 176 249 340 451 584 741 924 ...
1 5 15 36 75 141 245 400 621 925 1331 1860 2535 3381 ...
1 6 21 56 127 258 483 848 1413 2254 3465 5160 7475 10570 ...
1 7 28 84 210 463 931 1744 3087 5215 8470 13300 20280 30135 ...
1 8 36 120 330 792 1717 3440 6471 11560 19778 32616 52104 80952 ...
1 9 45 165 495 1287 3003 6436 12879 24355 43923 76077 127257 206493 ...
1 10 55 220 715 2002 5005 11440 24311 48630 92433 168180 294645 499422 ...
...
For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors. For A(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] + Binomial[d+1-n, n+1], {d, 1, 15}, {n, 1, d}] // Flatten
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CROSSREFS
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Cf. A325000 (unoriented), A325000(n,k-n) (chiral), A325001 (achiral), A325002 (exactly k colors), A327083 (edges, ridges), A337883 (faces, peaks), A325004 (orthotope facets, orthoplex vertices), A325012 (orthoplex facets, orthotope vertices).
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KEYWORD
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AUTHOR
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STATUS
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approved
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