A320643 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 3 colors (subsets).

0, 0, 0, 0, 0, 4, 12, 44, 137, 408, 1190, 3416, 9730, 27560, 78148, 221250, 627960, 1784038, 5081154, 14496956, 41455409, 118764600, 340919744, 980315700, 2823696150, 8145853520, 23533759241, 68081765650, 197206716570, 571906256808, 1660387879116, 4825525985408, 14037945170525, 40875277302720, 119122416494961, 347440682773324, 1014151818975190, 2962391932326680, 8659301777595196, 25328461701728194

Offset: 1

Author

Robert A. Russell, Oct 18 2018

Keywords

• nonn
• easy

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 A056296 and A304973, which can be used in conjunction with the first formula.

Examples

			For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
		

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.

Crossrefs

Column 3 of A320647.

Cf. A056296 (oriented), A056358 (unoriented), A304973 (achiral).

Programs

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=3; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]

Formula

a(n) = (A056296(n) - A304973(n)) / 2 = A056296(n) - A056358(n) = A056358(n) - A304973(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=3 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)).