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A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets). 3

%I #13 Nov 04 2019 14:52:41

%S 0,0,0,0,0,6,34,190,996,5070,26454,139484,749742,4082481,22509626,

%T 125231540,702004040,3958071545,22423227634,127524417922,727617119592,

%U 4163076477731,23876455868772,137228326265794,790200053665362,4557942281943078,26331297198477874,152331940294133402,882422871962784662,5117852332008063806,29715786649820358328,172720006045619486686,1004904748993330281274,5852047136464153694312

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

%C Two color patterns are equivalent if the colors are permuted.

%C Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.

%C There are nonrecursive formulas, generating functions, and computer programs for A056294 and A305752, which can be used in conjunction with the first formula.

%H Andrew Howroyd, <a href="/A320746/b320746.txt">Table of n, a(n) for n = 1..200</a>

%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.

%F a(n) = (A056294(n) - A305752(n)) / 2 = A056294(n) - A056356(n) = A056356(n) - A305752(n).

%F 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)).

%F a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n) + A320646(n).

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

%t 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]]

%t 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 *)

%t k=6; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

%Y Column 6 of A320742.

%Y Cf. A056294 (oriented), A056356 (unoriented), A305752 (achiral).

%K nonn,easy

%O 1,6

%A _Robert A. Russell_, Oct 21 2018

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