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For n>1, A337413(n,k) = Sum_{j=1..2*n*(n-1)} T(n,j) * binomial(k,j).

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T(n,k) = A338146(n,k) - A338147(n,k) = (A338146(n,k) - A338149(n,k)) / 2 = A338146A338147(n,k) - A338149(n,k).

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allocated for Robert ATriangle read by rows: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-D orthoplex (or ridges of a regular n-D orthotope) using exactly k colors. Row 1 has 1 column; row n>1 has 2*n*(n-1) columns. Russell

0, 0, 0, 3, 3, 0, 74, 10482, 303268, 3440700, 19842840, 65867760, 133580160, 168399000, 128898000, 54885600, 9979200, 0, 40927, 731157018, 729348051686, 151526009158620, 11418355290999750, 415756294427389020, 8643340000393019040

1,4

Chiral colorings come in pairs, each the reflection of the other. A ridge is an (n-2)-face of an n-D polytope. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges (vertices). For n=3, the figure is an octahedron (cube) with 12 edges. For n>1, the number of edges (ridges) is 2*n*(n-1). The Schläfli symbols for the n-D orthotope (hypercube) and the n-D orthoplex (hyperoctahedron, cross polytope) are {4,3,...,3,3} and {3,3,...,3,4} respectively, with n-2 3's in each case. The figures are mutually dual.

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).

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.

A337413(n,k) = Sum_{j=1..2*n} T(n,j) * binomial(k,j).

T(n,k) = A338146(n,k) - A338147(n,k) = (A338146(n,k) - A338149(n,k)) / 2 = A338146(n,k) - A338149(n,k).

T(2,k) = A338144(2,k) = A325018(2,k) = A325010(2,k); T(3,k) = A338144(3,k).

Triangle begins with T(1,1):

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0 74 10482 303268 3440700 19842840 65867760 133580160 168399000 ...

For T(2,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. For T(2,4)=3, the chiral pairs are ABCD-ADCB, ACBD-ADBC, and ABDC-ACDB.

m=1; (* dimension of color element, here an edge *)

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

Join[{{0}}, Table[LinearSolve[Table[Binomial[i, j], {i, 2^(m+1)Binomial[n, m+1]}, {j, 2^(m+1)Binomial[n, m+1]}], Table[array[n, k], {k, 2^(m+1)Binomial[n, m+1]}]], {n, m+1, m+4}]] // Flatten

Cf. A338146 (oriented), A338147 (unoriented), A338149 (achiral), A337413 (k or fewer colors), A325010 (orthoplex vertices, orthotope facets).

Cf. A327089 (simplex), A338144 (orthotope edges, orthoplex ridges).

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nonn,tabf

Robert A. Russell, Oct 12 2020

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allocated for Robert A. Russell

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