A327085 -id:A327085 - cp's OEIS frontend

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.

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

Offset: 1

Robert A. Russell, Aug 19 2019

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) is also the number of achiral colorings of the vertices (0-dimensional simplices) of an equilateral triangle.

			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.

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.

Cf. A327087 (oriented), A327088 (unoriented), A327089 (chiral), A327086 (up to k colors), A325003 (vertices).

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

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).

A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

Original entry on oeis.org

0, 1, 21, 140, 575, 1785, 4606, 10416, 21330, 40425, 71995, 121836, 197561, 308945, 468300, 690880, 995316, 1404081, 1943985, 2646700, 3549315, 4694921, 6133226, 7921200, 10123750, 12814425, 16076151, 20001996

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

Each member of a chiral pair is a reflection, but not a rotation, of the other. A regular tetrahedron has 6 edges and Schläfli symbol {3,3}.

			For a(2)=1, two opposite edges and one edge connecting those have one color; the other three edges have the other color.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A046023(unoriented), A063842(n-1) (oriented), A037270 (chiral).
Other elements: A000332 (vertices and faces).
Other polyhedra: A337406 (cube/octahedron).
Row 3 of A327085 (chiral pairs of colorings of edges or ridges of an n-simplex).

Mathematica
```
Table[(n-1)n^2(n+1)(n^2-2)/24, {n, 40}]
```

a(n) = (n-1) * n^2 * (n+1) * (n^2-2) / 24.
a(n) = 1*C(n,2) + 18*C(n,3) + 62*C(n,4) + 75*C(n,5) + 30*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A046023(n) - A063842(n-1) = (A046023(n) - A037270(n)) / 2 = A063842(n-1) - A037270(n).
G.f.: x^2 * (1+x) * (1+13x+x^2)/(1-x)^7.

cp's OEIS Frontend

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.

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