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A320646 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 6 colors (subsets). 3
0, 0, 0, 0, 0, 0, 0, 9, 125, 1054, 7928, 54383, 356594, 2259504, 14008733, 85422360, 514773336, 3074341497, 18238301412, 107649939612, 632987843336, 3711471738408, 21716706883190, 126879832615600, 740528154956264, 4319137675225128, 25181504728152534, 146788320134425736, 855660631677225738, 4988501691655508510, 29089896998939710698 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
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 A056299 and A304976, which can be used in conjunction with the first formula.
LINKS
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
FORMULA
a(n) = (A056299(n) - A304976(n)) / 2 = A056299(n) - A056361(n) = A056361(n) - A304976(n).
a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 is number of colors or sets, 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)).
EXAMPLE
For a(8)=9, the chiral pairs are AABACDEF-AABCDEAF, AABCADEF-AABCDAEF, AABCBDEF-AABCDEFE, AABCDBEF-AABCDEFD, AABCDEBF-AABCDEFC, AABCDCEF-AABCDEDF, ABACDEBF-ABACDEBF, ABCADBEF-ABCADECF, and ABCDAEBF-ABCADBEF.
MATHEMATICA
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 *)
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]]
k=6; Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n) - Ach[n, k]/2, {n, 40}]
CROSSREFS
Column 6 of A320647.
Cf. A056299 (oriented), A056361 (unoriented), A304976 (achiral).
Sequence in context: A280896 A364940 A138438 * A241709 A085528 A192724
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 19 2018
STATUS
approved

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Last modified March 28 11:58 EDT 2024. Contains 371241 sequences. (Running on oeis4.)