A337893 - OEIS

A337893

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

0, 0, 0, 0, 1, 0, 0, 66, 11158298, 0, 0, 920, 4825452718593, 314824333015938998688, 0, 0, 6350, 48038354542204960, 38491882659300767730994725249684096, 31716615393292685397985382790580028572676096, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

2,8

COMMENTS

Each member of a chiral pair is a reflection, but not a rotation, of the other. 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).

Also the number of chiral pairs of colorings of the peaks of an n-dimensional orthotope (hypercube). A peak is an (n-3)-dimensional orthotope.

LINKS

Table of n, a(n) for n=2..22.

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

FORMULA

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).

T(n,k) = A337891(n,k) - A337892(n,k) = (A337891(n,k) - A337894(n,k)) / 2 = A337892(n,k) - A337894(n,k).

EXAMPLE

Table begins with T(2,1):

0 0 0 0 0 ...

0 1 66 920 6350 ...

0 11158298 4825452718593 48038354542204960 60632976384183154375 ...

MATHEMATICA

m=2; (* dimension of color element, here a face *)

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]];

FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}]; DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);

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}]; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3, n}]], 1, -1]Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[]);

PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0, cs]]]);

pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)

row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^n)]

array[n_, k_] := row[n] /. b -> k

Table[array[n, d+m-n], {d, 6}, {n, m, d+m-1}] // Flatten

CROSSREFS

Cf. A337891 (oriented), A337892 (unoriented), A337894 (achiral).

Other elements: A325006 (vertices), A337413 (edges).

Other polytopes: A337885 (simplex), A337889 (orthotope).

Rows 2-4 are A000004, A337896, A331360.

Sequence in context: A358862 A295790 A246241 * A008991 A051323 A112506

Adjacent sequences: A337890 A337891 A337892 * A337894 A337895 A337896

KEYWORD

nonn,tabl

AUTHOR

Robert A. Russell, Sep 28 2020

STATUS

approved