login
A320743
Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).
4
0, 0, 0, 0, 0, 4, 13, 46, 144, 420, 1221, 3474, 9856, 27794, 78632, 222156, 629760, 1787440, 5087797, 14509580, 41479867, 118811286, 341009901, 980488510, 2824029648, 8146494860, 23534997912, 68084154502, 197211336576, 571915188840, 1660405181149, 4825559508106, 14038010213051, 40875403561680, 119122661856133, 347441159864556, 1014152747485696
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 A002076 and A182522, 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) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(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=3 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).
EXAMPLE
For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
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=3; Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, k}], {n, 40}]
CROSSREFS
Column 3 of A320742.
Cf. A002076 (oriented), A056353 (unoriented), A182522 (achiral).
Sequence in context: A155328 A096353 A034553 * A104460 A095128 A149433
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 21 2018
STATUS
approved