cp's OEIS Frontend

A337888 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the square faces of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

%I A337888 #4 Sep 28 2020 21:41:34
%S A337888 1,2,1,3,10,1,4,56,49127,1,5,220,740360358,314824532572147370464,1,6,
%T A337888 680,733776248840,38491882660671134164965704408524083,
%U A337888 38343035259947576596859948806931124970404417593861154473053467181056,1
%N A337888 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the square faces of a regular n-dimensional orthotope (hypercube) using k or fewer colors.
%C A337888 Each chiral pair is counted as one when enumerating unoriented arrangements. Each face is a square bounded by four edges. For n=2, the figure is a square with one face. For n=3, the figure is a cube with 6 faces. For n=4, the figure is a tesseract with 24 faces. The number of faces is 2^(n-2)*C(n,2).
%C A337888 Also the number of unoriented colorings of peaks of an n-dimensional orthoplex. A peak is an (n-3)-dimensional simplex.
%H A337888 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 A337888 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 A337888 T(n,k) = A337887(n,k) - A337889(n,k) = (A337887(n,k) + A337890(n,k)) / 2 = A337889(n,k) + A337890(n,k).
%e A337888 Array begins with T(2,1):
%e A337888  1     2         3            4               5                 6 ...
%e A337888  1    10        56          220             680              1771 ...
%e A337888  1 49127 740360358 733776248840 155261523065875 12340612271439081 ...
%t A337888 m=2; (* dimension of color element, here a square face *)
%t A337888 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, n - m]];
%t A337888 FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
%t A337888 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 A337888 PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
%t A337888 pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
%t A337888 row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^n)]
%t A337888 array[n_, k_] := row[n] /. b -> k
%t A337888 Table[array[n,d+m-n], {d,6}, {n,m,d+m-1}] // Flatten
%Y A337888 Cf. A337887 (oriented), A337889 (chiral), A337890 (achiral).
%Y A337888 Other elements: A325013 (vertices), A337408 (edges).
%Y A337888 Other polytopes: A337884 (simplex), A337892 (orthoplex).
%Y A337888 Rows 2-4 are A000027, A198833, A331355.
%K A337888 tabl,nonn
%O A337888 2,2
%A A337888 _Robert A. Russell_, Sep 28 2020