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%I #26 Aug 27 2020 04:32:18
%S 1,2,1,3,6,1,4,18,21,1,5,40,201,308,1,6,75,1076,34128,180342,1,7,126,
%T 4025,1056576,2945136213,366975285216,1,8,196,11901,15303750,
%U 2932338749408,103863386269870076808,10316179427644325573474464,1
%N Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using up to k colors.
%C Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. An achiral coloring is identical to its reflection.
%C Also the number of achiral colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.
%H Robert A. Russell, <a href="/A325015/b325015.txt">Table of n, a(n) for n = 1..78</a>
%H E. M. Palmer and R. W. Robinson, <a href="https://doi.org/10.1007/BF02392038">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%F The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
%F T(n,k) = 2*A325013(n,k) - A325012(n,k) = A325012(n,k) - 2*A325014(n,k) = A325013(n,k) - A325014(n,k).
%F T(n,k) = Sum_{j=1..3*2^(n-2)} A325019(n,j) * binomial(k,j).
%e Array begins with T(1,1):
%e 1 2 3 4 5 6 7 8 ...
%e 1 6 18 40 75 126 196 288 ...
%e 1 21 201 1076 4025 11901 29841 66256 ...
%e 1 308 34128 1056576 15303750 136236276 865711763 4296782848 ...
%e ...
%e For T(2,2)=6, two squares have all edges the same color, two have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.
%t a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
%t a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
%t CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
%t CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
%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 cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
%t Unprotect[Times]; Times[CI[a_List], CI[b_List]] := (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
%t CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
%t CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
%t pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t row[n_Integer] := row[n] = Factor[(Total[(CI1[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
%t array[n_, k_] := row[n] /. j -> k
%t Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten
%Y Cf. A325012 (oriented), A325013 (unoriented), A325014 (chiral), A325019 (exactly k colors).
%Y Other n-dimensional polytopes: A325001 (simplex), A325007 (orthotope).
%Y Rows 1-2 are A000027, A002411.
%Y Cf. A000048, A001037.
%K nonn,tabl,easy
%O 1,2
%A _Robert A. Russell_, May 27 2019
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