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A337412 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

%I A337412 #15 Aug 28 2020 06:05:20
%S A337412 1,2,1,3,6,1,4,21,144,1,5,55,12111,49127,1,6,120,358120,740360358,
%T A337412 293122232,1,7,231,5131650,733776248840,3168520600399659,
%U A337412 25174334733080,1,8,406,45528756,155261523065875,314848558732420555904,920040738175691418086226,30035285091978202824,1
%N A337412 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.
%C A337412 Each chiral pair is counted as one when enumerating unoriented arrangements. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is an octahedron with 12 edges. The number of edges is 2n*(n-1) for n>1.
%C A337412 Also the number of unoriented colorings of the regular (n-2)-dimensional orthotopes (hypercubes) in a regular n-dimensional orthotope.
%H A337412 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 A337412 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 A337412 T(n,k) = A337411(n,k) - A337413(n,k) = (A337411(n,k) + A337414(n,k)) / 2 = A337413(n,k) + A337414(n,k).
%e A337412 Table begins with T(1,1):
%e A337412 1   2     3      4       5        6         7          8          9 ...
%e A337412 1   6    21     55     120      231       406        666       1035 ...
%e A337412 1 144 12111 358120 5131650 45528756 288936634 1433251296 5887880415 ...
%e A337412 For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.
%t A337412 m=1; (* dimension of color element, here an edge *)
%t A337412 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 A337412 FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
%t A337412 CCPol[r_List] := (r1 = r; r2 = cs - r1; 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 A337412 PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
%t A337412 pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
%t A337412 row[m]=b;
%t A337412 row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^n)]
%t A337412 array[n_, k_] := row[n] /. b -> k
%t A337412 Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten
%Y A337412 Cf. A337411 (oriented), A337413 (chiral), A337414 (achiral).
%Y A337412 Rows 1-4 are A000027, A002817, A199406, A331355.
%Y A337412 Cf. A327084 (simplex edges), A337408 (orthotope edges), A325005 (orthoplex vertices).
%K A337412 nonn,tabl
%O A337412 1,2
%A A337412 _Robert A. Russell_, Aug 26 2020