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%I #21 Jun 09 2021 02:31:38
%S 0,0,0,0,0,0,0,1,1,0,0,4,21,6,0,0,10,140,405,28,0,0,20,575,7904,17154,
%T 252,0,0,35,1785,76880,1415648,1920375,4726,0,0,56,4606,486522,
%U 41453650,855834880,547375212,150324,0
%N Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional simplex using up to k colors.
%C 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. The chiral colorings of its edges come in pairs, each the reflection of the other.
%C A(n,k) is also the number of chiral pairs of 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 chiral pairs of colorings of the vertices (0-dimensional simplices) of an equilateral triangle.
%H Robert A. Russell, <a href="/A327085/b327085.txt">Table of n, a(n) for n = 1..325</a> First 25 antidiagonals.
%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00_42.html">The cycle type of the induced action on 2-subsets</a>
%H E. M. Palmer and R. W. Robinson, <a href="https://projecteuclid.org/euclid.acta/1485889789">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%F 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.
%F A(n,k) = Sum_{j=1..(n+1)*n/2} A327089(n,j) * binomial(k,j).
%F A(n,k) = A327083(n,k) - A327084(n,k) = (A327083(n,k) - A327086(n,k)) / 2 = A327084(n,k) - A327086(n,k).
%e Array begins with A(1,1):
%e 0 0 0 0 0 0 0 0 0 0 0 ...
%e 0 0 1 4 10 20 35 56 84 120 165 ...
%e 0 1 21 140 575 1785 4606 10416 21330 40425 71995 ...
%e 0 6 405 7904 76880 486522 2300305 8806336 28725192 82626270 214744629 ...
%e ...
%e For A(2,3) = 1, the chiral pair is ABC-ACB.
%t CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)
%t CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]
%t 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)
%t 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]]]
%t pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#,2]]], 1, -1] pc[#] j^Total[CycleX[#]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
%t array[n_, k_] := row[n] /. j -> k
%t Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten
%t (* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)
%t pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
%t ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]
%t array[n_,k_] := Total[If[EvenQ[Total[1-Mod[#,2]]],1,-1] pc[#]k^ex[#] &/@ IntegerPartitions[n+1]]/(n+1)!
%t Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten
%Y Cf. A327083 (oriented), A327084 (unoriented), A327086 (achiral), A327089 (exactly k colors), A325000(n,k-n) (vertices, facets), A337885 (faces, peaks), A337409 (orthotope edges, orthoplex ridges), A337413 (orthoplex edges, orthotope ridges).
%Y Rows 1-4 are A000004, A000292(n-2), A337899, A331352.
%K nonn,tabl
%O 1,12
%A _Robert A. Russell_, Aug 19 2019
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