A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).

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

Offset

1,6

Comments

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.

Links

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.

Formula

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).

Example

For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Mathematica

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}]

Crossrefs

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

Keyword

nonn,easy

Author

Robert A. Russell, Oct 21 2018

Status

approved