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A325018 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

%I A325018 #5 Jun 11 2019 06:09:44
%S A325018 0,1,0,0,3,3,0,1,63,662,2400,3900,2940,840,0,94,97692,10308758,
%T A325018 337560150,5098740090,42976836210,224685801060,775389028050,
%U A325018 1830791421900,3007909258200,3439214024400,2685727044000,1366701336000,408648240000,54486432000
%N A325018 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.
%C A325018 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. The chiral colorings of its facets come in pairs, each the reflection of the other.
%C A325018 Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthotope (cube) using exactly k colors.
%H A325018 Robert A. Russell, <a href="/A325018/b325018.txt">Table of n, a(n) for n = 1..510</a>, rows 1..8, flattened.
%H A325018 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 A325018 A325014(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).
%F A325018 T(n,k) = A325016(n,k) - A325017(n,k) = (A325016(n,k) - A325019(n,k)) / 2 = A325017(n,k) - A325019(n,k).
%e A325018 Triangle begins with T(1,1):
%e A325018 0 1
%e A325018 0 0  3   3
%e A325018 0 1 63 662 2400 3900 2940 840
%e A325018 For T(2,3)=3, each square has one of the three colors on two adjacent edges.
%t A325018 a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&,n,EvenQ],MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
%t A325018 a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
%t A325018 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 A325018 CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, (a37 /@ sub)/2}]]] 2^(n-1); (* odd perm. *)
%t A325018 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 A325018 cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
%t A325018 Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
%t A325018 CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
%t A325018 CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
%t A325018 pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A325018 row[n_Integer] := row[n] = Factor[(Total[((CI0[#] - CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
%t A325018 array[n_, k_] := row[n] /. j -> k (* A325014 *)
%t A325018 Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten
%Y A325018 Cf. A325016 (oriented), A325017 (unoriented), A325019 (achiral), A325014 (up to k colors).
%Y A325018 Other n-dimensional polytopes: A325010 (orthotope).
%Y A325018 Cf. A000048, A001037.
%K A325018 nonn,tabf
%O A325018 1,5
%A A325018 _Robert A. Russell_, Jun 09 2019