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A320746
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Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).
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3
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0, 0, 0, 0, 0, 6, 34, 190, 996, 5070, 26454, 139484, 749742, 4082481, 22509626, 125231540, 702004040, 3958071545, 22423227634, 127524417922, 727617119592, 4163076477731, 23876455868772, 137228326265794, 790200053665362, 4557942281943078, 26331297198477874, 152331940294133402, 882422871962784662, 5117852332008063806, 29715786649820358328, 172720006045619486686, 1004904748993330281274, 5852047136464153694312
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OFFSET
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1,6
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COMMENTS
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Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056294 and A305752, which can be used in conjunction with the first formula.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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FORMULA
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a(n) = (A056294(n) - A305752(n)) / 2 = A056294(n) - A056356(n) = A056356(n) - A305752(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=6 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n) + A320646(n).
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EXAMPLE
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For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.
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MATHEMATICA
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Adnk[d_, n_, k_] := Adnk[d, n, k] = If[n>0 && k>0, Adnk[d, n-1, k]k + DivisorSum[d, Adnk[d, n-1, k-#]&], Boole[n == 0 && k == 0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *)
k=6; Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, k}], {n, 40}]
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CROSSREFS
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Column 6 of A320742.
Cf. A056294 (oriented), A056356 (unoriented), A305752 (achiral).
Sequence in context: A085351 A125343 A163350 * A320749 A052264 A049608
Adjacent sequences: A320743 A320744 A320745 * A320747 A320748 A320749
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KEYWORD
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nonn,easy
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AUTHOR
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Robert A. Russell, Oct 21 2018
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STATUS
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approved
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