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A325012 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

%I A325012 #29 Jun 15 2019 14:43:14
%S A325012 1,4,1,9,6,1,16,24,23,1,25,70,333,496,1,36,165,2916,230076,2275974,1,
%T A325012 49,336,16725,22456756,965227578201,800648638402240,1,64,616,70911,
%U A325012 795467350,9607713956430560,149031415906337877339236058,1054942853799126580390222487977120,1
%N A325012 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.
%C A325012 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 oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
%C A325012 Also the number of oriented colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.
%H A325012 Robert A. Russell, <a href="/A325012/b325012.txt">Table of n, a(n) for n = 1..78</a>
%H A325012 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 A325012 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 A325012 A(n,k) = A325013(n,k) + A325014(n,k) = 2*A325013(n,k) - A325015(n,k) = 2*A325014(n,k) + A325015(n,k).
%F A325012 A(n,k) = Sum_{j=1..2^n} A325016(n,j) * binomial(k,j).
%e A325012 Array begins with A(1,1):
%e A325012 1   4      9       16        25          36           49            64 ...
%e A325012 1   6     24       70       165         336          616          1044 ...
%e A325012 1  23    333     2916     16725       70911       241913        701968 ...
%e A325012 1 496 230076 22456756 795467350 14697611496 173107727191 1466088119056 ...
%e A325012 For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses one color for each vertex.
%t A325012 a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&,n,EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
%t A325012 a37[n_] := a37[n] = DivisorSum[n,MoebiusMu[n/#]2^#&]/n; (* A001037 *)
%t A325012 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 A325012 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 A325012 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 A325012 cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
%t A325012 Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
%t A325012 CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
%t A325012 CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
%t A325012 pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A325012 row[n_Integer] := row[n] = Factor[(Total[(CI0[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
%t A325012 array[n_, k_] := row[n] /. j -> k
%t A325012 Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten
%Y A325012 Cf. A325013 (unoriented), A325014 (chiral), A325015 (achiral), A325016 (exactly k colors).
%Y A325012 Other n-dimensional polytopes: A324999 (simplex), A325004 (orthotope).
%Y A325012 Rows 1-3 are A000290, A006528, A000543; column 2 is A237748.
%Y A325012 Cf. A000048, A001037.
%K A325012 nonn,tabl,easy
%O A325012 1,2
%A A325012 _Robert A. Russell_, May 27 2019