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A337894 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

%I A337894 #4 Sep 28 2020 21:42:29
%S A337894 1,2,1,3,21,1,4,201,93024,1,5,1076,294157089,199556208371776,1,6,4025,
%T A337894 91983927296,1370366433970979158839987,
%U A337894 346179533768149850758531729588224,1
%N A337894 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.
%C A337894 An achiral arrangement is identical to its reflection. For n=2, the figure is a square with one square face. For n=3, the figure is an octahedron with 8 triangular faces. For higher n, the number of triangular faces is 8*C(n,3).
%C A337894 Also the number of achiral colorings of the peaks of an n-dimensional orthotope (hypercube). A peak is an (n-3)-dimensional orthotope.
%H A337894 K. Balasubramanian, <a href="https://doi.org/10.33187/jmsm.471940">Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications</a>, J. Math. Sci. & Mod. 1 (2018), 158-180.
%F A337894 The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).
%F A337894 T(n,k) = 2*A337892(n,k) - A337891(n,k) = A337891(n,k) - 2*A337893(n,k) = A337892(n,k) - A337893(n,k).
%e A337894 Table begins with T(2,1):
%e A337894 1     2         3           4             5               6 ...
%e A337894 1    21       201        1076          4025           11901 ...
%e A337894 1 93024 294157089 91983927296 7960001890625 304914963625056 ...
%t A337894 m=2; (* dimension of color element, here a face *)
%t A337894 Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, m+1]];
%t A337894 FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
%t A337894 CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],0,(per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[])]);
%t A337894 PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
%t A337894 pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
%t A337894 row[m]=b;
%t A337894 row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
%t A337894 array[n_, k_] := row[n] /. b -> k
%t A337894 Table[array[n,d+m-n], {d,7}, {n,m,d+m-1}] // Flatten
%Y A337894 Cf. A337891 (oriented), A337892 (unoriented), A337893 (chiral).
%Y A337894 Other elements: A325007 (vertices), A337414 (edges).
%Y A337894 Other polytopes: A337886 (simplex), A337890 (orthotope).
%Y A337894 Rows 2-4 are A000027, A337897, A331361.
%K A337894 nonn,tabl
%O A337894 2,2
%A A337894 _Robert A. Russell_, Sep 28 2020