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A327090 Triangle read by rows: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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. An achiral coloring is identical to its reflection.

T(n,k) is also the number of achiral colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using exactly k colors. Thus, T(2,k) also counts the number of achiral colorings of the vertices (0-dimensional 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), 123-143.

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 n-1 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, (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]]], 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

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Last modified April 11 22:15 EDT 2021. Contains 342892 sequences. (Running on oeis4.)