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A327084
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Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the edges of a regular n-dimensional simplex using up to k colors.
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14
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1, 2, 1, 3, 4, 1, 4, 10, 11, 1, 5, 20, 66, 34, 1, 6, 35, 276, 792, 156, 1, 7, 56, 900, 10688, 25506, 1044, 1, 8, 84, 2451, 90005, 1601952, 2302938, 12346, 1, 9, 120, 5831, 533358, 43571400, 892341888, 591901884, 274668
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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 unoriented colorings are the same if congruent; chiral pairs are counted as one.
A(n,k) is also the number of unoriented 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 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.
A(n,k) = Sum_{j=1..(n+1)*n/2} A327088(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 10 20 35 56 84 120 165 220 ...
1 11 66 276 900 2451 5831 12496 24651 45475 ...
1 34 792 10688 90005 533358 2437848 9156288 29522961 84293770 ...
...
For A(2,3) = 10, 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[pc[#] j^Total[CycleX[#]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
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 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[pc[#]k^ex[#] &/@ IntegerPartitions[n+1]]/(n+1)!
Table[array[n, d-n+1], {d, 10}, {n, d}] // Flatten
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total[Quotient[v, 2]];
T[n_, k_] := Module[{s = 0}, Do[s += permcount[p]*k^edges[p], {p, IntegerPartitions[n+1]}]; s/(n+1)!];
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PROG
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(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
T(n, k) = {my(s=0); forpart(p=n+1, s+=permcount(p)*k^edges(p)); s/(n+1)!} \\ Andrew Howroyd, Sep 06 2019
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CROSSREFS
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Cf. A327083 (oriented), A327085 (chiral), A327086 (achiral), A327088 (exactly k colors), A325000 (vertices, facets), A337884 (faces, peaks), A337408 (orthotope edges, orthoplex ridges), A337412 (orthoplex edges, orthotope ridges).
Cf. A063841 (k-multigraphs on n nodes).
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KEYWORD
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AUTHOR
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STATUS
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approved
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