

A327090


Triangle read by rows: T(n,k) is the number of achiral colorings of the edges of a regular ndimensional simplex using exactly k colors. Row n has (n+1)*n/2 columns.


8



1, 1, 2, 0, 1, 8, 18, 12, 0, 0, 1, 26, 306, 1400, 2800, 2520, 840, 0, 0, 0, 1, 126, 7971, 153660, 1268475, 5463990, 13534290, 20018880, 17478720, 8316000, 1663200, 0, 0, 0, 0
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OFFSET

1,3


COMMENTS

An ndimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The SchlĂ¤fli symbol, {3,...,3}, of the regular ndimensional simplex consists of n1 threes. An achiral coloring is identical to its reflection.
T(n,k) is also the number of achiral colorings of (n2)dimensional regular simplices in an ndimensional simplex using exactly k colors. Thus, T(2,k) also counts the number of achiral colorings of the vertices (0dimensional simplices) of an equilateral triangle.


LINKS

Robert A. Russell, Table of n, a(n) for n = 1..220 First 10 rows.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123143.


FORMULA

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. The last n1 entries in row n are zero.
A327086(n,k) = Sum_{j=1..(n+1)*n/2} T(n,j) * binomial(k,j).
A(n,k) = 2*A327084(n,k)  A327083(n,k) = A327083(n,k)  2*A327085(n,k) = A327084(n,k)  A327085(n,k).


EXAMPLE

Triangle begins with T(1,1):
1
1 2 0
1 8 18 12 0 0
1 26 306 1400 2800 2520 840 0 0 0
For T(2,2) = 2, the two colorings of the triangle edges are AAB and ABB.


MATHEMATICA

CycleX[{2}] = {{1, 1}}; (* cycle index for permutation with given cycle structure *)
CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2, 1}, {n, (n2)/2}}, {{n, (n1)/2}}]
compress[x : {{_, _} ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i = 1, If[s[[i, 1]] == s[[i1, 1]], s[[i1, 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[1Mod[#, 2]]], 0, pc[#] j^Total[CycleX[#]][[2]]] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
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


CROSSREFS

Cf. A327087 (oriented), A327088 (unoriented), A327089 (chiral), A327086 (up to k colors), A325003 (vertices).
Sequence in context: A055140 A335330 A191936 * A021836 A255306 A072551
Adjacent sequences: A327087 A327088 A327089 * A327091 A327092 A327093


KEYWORD

nonn,tabf


AUTHOR

Robert A. Russell, Aug 19 2019


STATUS

approved



