cp's OEIS Frontend

A337891 Array read by descending antidiagonals: T(n,k) is the number of oriented colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

%I A337891 #10 Sep 29 2020 10:05:34
%S A337891 1,2,1,3,23,1,4,333,22409620,1,5,2916,9651199594275,
%T A337891 629648865588086369152,1,6,16725,96076801068337216,
%U A337891 76983765319971901895960429658208179,63433230786931550329738915431918588874940416,1
%N A337891 Array read by descending antidiagonals: T(n,k) is the number of oriented colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.
%C A337891 Each chiral pair is counted as two when enumerating oriented arrangements. 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 A337891 Also the number of oriented colorings of the peaks of an n-dimensional orthotope (hypercube). A peak is an (n-3)-dimensional orthotope.
%C A337891 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).
%H A337891 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 A337891 T(n,k) = A337892(n,k) + A337893(n,k) = 2*A337892(n,k) - A337894(n,k) = 2*A337893(n,k) + A337894(n,k).
%e A337891 Array begins with T(2,1):
%e A337891 1        2             3                 4                     5 ...
%e A337891 1       23           333              2916                 16725 ...
%e A337891 1 22409620 9651199594275 96076801068337216 121265960728368199375 ...
%t A337891 m=2; (* dimension of color element, here a face *)
%t A337891 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 A337891 FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
%t A337891 CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]], (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[]),0]);
%t A337891 PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
%t A337891 pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
%t A337891 row[m]=b;
%t A337891 row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
%t A337891 array[n_, k_] := row[n] /. b -> k
%t A337891 Table[array[n,d+m-n], {d,6}, {n,m,d+m-1}] // Flatten
%Y A337891 Cf. A337892 (unoriented), A337893 (chiral), A337894 (achiral).
%Y A337891 Other elements: A325004 (vertices), A337411 (edges).
%Y A337891 Other polytopes: A337883 (simplex), A337887 (orthotope).
%Y A337891 Rows 2-4 are A000027, A000543, A331358
%K A337891 nonn,tabl
%O A337891 2,2
%A A337891 _Robert A. Russell_, Sep 28 2020