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A327083
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Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the edges of a regular n-dimensional simplex using up to k colors.
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13
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1, 2, 1, 3, 4, 1, 4, 11, 12, 1, 5, 24, 87, 40, 1, 6, 45, 416, 1197, 184, 1, 7, 76, 1475, 18592, 42660, 1296, 1, 8, 119, 4236, 166885, 3017600, 4223313, 17072, 1, 9, 176, 10437, 1019880, 85025050, 1748176768, 1139277096, 424992
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OFFSET
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1,2
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COMMENTS
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An n-dimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n-2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
A(n,k) is also the number of oriented colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) is also the number of oriented colorings of the vertices (0-dimensional simplices) of an equilateral triangle.
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LINKS
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FORMULA
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The algorithm used in the Mathematica program below assigns each permutation of the vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = Sum_{j=1..(n+1)*n/2} A327087(n,j) * binomial(k,j).
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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 4 11 24 45 76 119 176 249 340 ...
1 12 87 416 1475 4236 10437 22912 45981 85900 ...
1 40 1197 18592 166885 1019880 4738153 17962624 58248153 166920040 ...
...
For A(2,3) = 11, the nine achiral colorings are AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, and CCC. The chiral pair is ABC-ACB.
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MATHEMATICA
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CycleX[{2}] = {{1, 1}}; (* cycle index for permutation with given cycle structure *)
CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2, 1}, {n, (n-2)/2}}, {{n, (n-1)/2}}]
compress[x : {{_, _} ...}] := (s = Sort[x]; For[i=Length[s], i>1, i-=1, If[s[[i, 1]] == s[[i-1, 1]], s[[i-1, 2]]+=s[[i, 2]]; s=Delete[s, i], Null]]; s)
CycleX[p_List] := CycleX[p] = compress[Join[CycleX[Drop[p, -1]], If[Last[p] > 1, CycleX[{Last[p]}], ## &[]], If[# == Last[p], {#, Last[p]}, {LCM[#, Last[p]], GCD[#, Last[p]]}] & /@ Drop[p, -1]]]
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (*partition count*)
row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#, 2]]], pc[#] j^Total[CycleX[#]][[2]], 0] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten
(* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)
pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, i-1}] + Total[Quotient[v, 2]]
array[n_, k_] := Total[If[EvenQ[Total[1-Mod[#, 2]]], pc[#]k^ex[#], 0] &/@ IntegerPartitions[n+1]]/((n+1)!/2)
Table[array[n, d-n+1], {d, 10}, {n, d}] // Flatten
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CROSSREFS
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Cf. A327084 (unoriented), A327085 (chiral), A327086 (achiral), A327087 (exactly k colors), A324999 (vertices, facets), A337883 (faces, peaks), A337407 (orthotope edges, orthoplex ridges), A337411 (orthoplex edges, orthotope ridges).
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KEYWORD
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AUTHOR
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STATUS
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approved
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