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%I #18 Jun 15 2019 14:43:32
%S 1,3,1,6,6,1,10,21,22,1,15,55,267,402,1,21,120,1996,132102,1228158,1,
%T 28,231,10375,11756666,484086357207,400507806843728,1,36,406,41406,
%U 405385550,4805323147589984,74515759884862073604656433,527471432057653004017274030725792,1
%N Array read by descending antidiagonals: A(n,k) is the number of unoriented 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. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.
%C Also the number of unoriented colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.
%H Robert A. Russell, <a href="/A325013/b325013.txt">Table of n, a(n) for n = 1..78</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.
%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 A(n,k) = A325012(n,k) - A325014(n,k) = (A325012(n,k) + A325015(n,k)) / 2 = A325014(n,k) + A325015(n,k).
%F A(n,k) = Sum_{j=1..2^n} A325017(n,j) * binomial(k,j).
%e Array begins with A(1,1):
%e 1 3 6 10 15 21 28 36 ...
%e 1 6 21 55 120 231 406 666 ...
%e 1 22 267 1996 10375 41406 135877 384112 ...
%e 1 402 132102 11756666 405385550 7416923886 86986719477 735192450952 ...
%e For A(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[((CI0[#] + CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. 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), A325014 (chiral), A325015 (achiral), A325017 (exactly k colors).
%Y Other n-dimensional polytopes: A325000 (simplex), A325005 (orthotope).
%Y Rows 1-4 are A000217, A002817, A128766, A128767; column 2 is A000616.
%Y Cf. A000048, A001037.
%K nonn,tabl,easy
%O 1,2
%A _Robert A. Russell_, May 27 2019
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