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 A325015 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. 11
 1, 2, 1, 3, 6, 1, 4, 18, 21, 1, 5, 40, 201, 308, 1, 6, 75, 1076, 34128, 180342, 1, 7, 126, 4025, 1056576, 2945136213, 366975285216, 1, 8, 196, 11901, 15303750, 2932338749408, 103863386269870076808, 10316179427644325573474464, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 SchlĂ¤fli 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. Also the number of achiral colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors. LINKS Robert A. Russell, Table of n, a(n) for n = 1..78 E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143. Wikipedia, Cross-polytope FORMULA 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. T(n,k) = 2*A325013(n,k) - A325012(n,k) = A325012(n,k) - 2*A325014(n,k) = A325013(n,k) - A325014(n,k). T(n,k) = Sum_{j=1..3*2^(n-2)} A325019(n,j) * binomial(k,j). EXAMPLE Array begins with T(1,1): 1   2     3       4        5         6         7          8 ... 1   6    18      40       75       126       196        288 ... 1  21   201    1076     4025     11901     29841      66256 ... 1 308 34128 1056576 15303750 136236276 865711763 4296782848 ... ... 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. MATHEMATICA a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *) a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *) 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. *) 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. *) 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) cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]}; Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times]; CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]] CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]] pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *) row[n_Integer] := row[n] = Factor[(Total[(CI1[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]]) array[n_, k_] := row[n] /. j -> k Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten CROSSREFS Cf. A325012 (oriented), A325013 (unoriented), A325014 (chiral), A325019 (exactly k colors). Other n-dimensional polytopes: A325001 (simplex), A325007 (orthotope). Rows 1-2 are A000027, A002411. Cf. A000048, A001037. Sequence in context: A115597 A325007 A103371 * A337414 A337410 A337389 Adjacent sequences:  A325012 A325013 A325014 * A325016 A325017 A325018 KEYWORD nonn,tabl,easy AUTHOR Robert A. Russell, May 27 2019 STATUS approved

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Last modified June 18 19:41 EDT 2021. Contains 345120 sequences. (Running on oeis4.)