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 A337409 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthotope (hypercube) using k or fewer colors. 9
 0, 0, 0, 0, 0, 0, 0, 3, 74, 0, 0, 15, 10704, 11158298, 0, 0, 45, 345640, 4825452718593, 314824408633217132928, 0, 0, 105, 5062600, 48038354542204960, 38491882659952177472606694634030116, 136221825854745676076981182469325427379054390050209792, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Each member of a chiral pair is a reflection, but not a rotation, of the other. 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 a cube with 12 edges. The number of edges is n*2^(n-1). Also the number of chiral pairs of colorings of the regular (n-2)-dimensional simplexes in a regular n-dimensional orthoplex. LINKS Table of n, a(n) for n=1..28. 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) = A337407(n,k) - A337408(n,k) = (A337407(n,k) - A337410(n,k)) / 2 = A337408(n,k) - A337410(n,k). EXAMPLE Table begins with T(1,1): 0 0 0 0 0 0 0 0 0 ... 0 0 3 15 45 105 210 378 630 ... 0 74 10704 345640 5062600 45246810 288005144 1430618784 5881281480 ... For T(2,3)=3, the chiral arrangements are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. MATHEMATICA 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+2x^(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]]; 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, 7}, {n, m, d+m-1}] // Flatten CROSSREFS Cf. A337407 (oriented), A337408 (unoriented), A337410 (achiral). Rows 2-4 are A050534, A337406, A331360. Cf. A327085 (simplex edges), A337413 (orthoplex edges), A325014 (orthotope vertices). Sequence in context: A336873 A121981 A337413 * A215961 A231780 A093183 Adjacent sequences: A337406 A337407 A337408 * A337410 A337411 A337412 KEYWORD nonn,tabl AUTHOR Robert A. Russell, Aug 26 2020 STATUS approved

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Last modified September 11 06:30 EDT 2024. Contains 375814 sequences. (Running on oeis4.)