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A327088
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Triangle read by rows: T(n,k) is the number of unoriented colorings of the edges of a regular n-dimensional simplex using exactly k colors. Row n has (n+1)*n/2 columns.
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8
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1, 1, 2, 1, 1, 9, 36, 74, 75, 30, 1, 32, 693, 7720, 44150, 138312, 247380, 252000, 136080, 30240, 1, 154, 25041, 1500860, 35815145, 423545178, 2868102286, 12141259200, 33841521000, 63823914000, 82023091920, 70832361600, 39351312000, 12713500800, 1816214400
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OFFSET
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1,3
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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 unoriented colorings are the same if congruent; chiral pairs are counted as one.
T(n,k) is also the number of unoriented colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using exactly k colors. Thus, T(2,k) is also the number of unoriented 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.
A327084(n,k) = Sum_{j=1..(n+1)*n/2} T(n,j) * binomial(k,j).
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EXAMPLE
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Triangle begins with T(1,1):
1
1 2 1
1 9 36 74 75 30
1 32 693 7720 44150 138312 247380 252000 136080 30240
For T(2,1)=1, all edges of the triangle are the same color. For T(2,2)=2, the edges are AAB and ABB. For T(2,3)=2, 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[pc[#] j^Total[CycleX[#]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
array[n_, k_] := row[n] /. j -> k
Table[LinearSolve[Table[Binomial[i, j], {i, 1, (n+1)n/2}, {j, 1, (n+1)n/2}], Table[array[n, k], {k, 1, (n+1)n/2}]], {n, 1, 6}] // Flatten
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CROSSREFS
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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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