A320749 Number of chiral pairs of color patterns (set partitions) in a cycle of length n.

0, 0, 0, 0, 0, 6, 34, 190, 1011, 5352, 29740, 172466, 1055232, 6793791, 46034940, 327303819, 2436650368, 18944771253, 153488081102, 1293086505784, 11306373089104, 102425178180769, 959825673145688, 9290807818971900, 92771800581171418, 954447025978145744, 10105871186441842623, 110009631951698573068, 1229996584263621368224, 14112483571723367245825, 166021918475962174194914, 2001010469483653602192695

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.

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) = Sum_{j=1..n} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where 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) = (A084423(n) - A080107(n)) / 2 = A084423(n) - A084708(n) = A084708(n) - A080107(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]]
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, n}], {n, 40}]

Crossrefs

Row sums of A320647.

Columns of A320742 converge to this as k increases.

Cf. A084423 (oriented), A084708 (unoriented), A080107 (achiral).

Sequence in context: A125343 A163350 A320746 * A052264 A049608 A244937

Adjacent sequences: A320746 A320747 A320748 * A320750 A320751 A320752

Keyword

nonn,easy

Author

Robert A. Russell, Oct 22 2018

Status

approved