A056299 -id:A056299 - cp's OEIS frontend

A152175 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations.

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 18, 13, 3, 1, 1, 9, 43, 50, 20, 3, 1, 1, 19, 126, 221, 136, 36, 4, 1, 1, 29, 339, 866, 773, 296, 52, 4, 1, 1, 55, 946, 3437, 4281, 2303, 596, 78, 5, 1, 1, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 1, 179, 7254, 51075, 115100, 110462, 52376, 13299, 1873, 147, 6, 1

Offset: 1

Vladeta Jovovic, Nov 27 2008

Number of n-bead necklace structures using exactly k different colored beads. Turning over the necklace is not allowed. Permuting the colors does not change the structure. - Andrew Howroyd, Apr 06 2017

			Triangle begins with T(1,1):
  1;
  1,   1;
  1,   1,     1;
  1,   3,     2,      1;
  1,   3,     5,      2,      1;
  1,   7,    18,     13,      3,      1;
  1,   9,    43,     50,     20,      3,      1;
  1,  19,   126,    221,    136,     36,      4,      1;
  1,  29,   339,    866,    773,    296,     52,      4,     1;
  1,  55,   946,   3437,   4281,   2303,    596,     78,     5,    1;
  1,  93,  2591,  13250,  22430,  16317,   5817,   1080,   105   , 5,   1;
  1, 179,  7254,  51075, 115100, 110462,  52376,  13299,  1873,  147,   6, 1;
  1, 315, 20125, 194810, 577577, 717024, 439648, 146124, 27654, 3025, 187, 6, 1;
  ...
For T(4,2)=3, the set partitions are AAAB, AABB, and ABAB.
For T(4,3)=2, the set partitions are AABC and ABAC.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Tilman Piesk, Partition related number triangles

Columns 2-6 are A056295, A056296, A056297, A056298, A056299.
Row sums are A084423.
Partial row sums include A000013, A002076, A056292, A056293, A056294.
Cf. A075195, A087854, A008277 (set partitions), A284949 (up to reflection), A152176 (up to rotation and reflection).
A(1,n,k) in formula is the Stirling subset number A008277.
A(2,n,k) in formula is A293181; A(3,n,k) in formula is A294201.

(* Using recursion formula from Gilbert and Riordan:*)
Adn[d_, n_] := Adn[d, n] = Which[0==n, 1, 1==n, DivisorSum[d, x^# &],
  1==d, Sum[StirlingS2[n, k] x^k, {k, 0, n}],
  True, Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n - 1], x] x]];
Table[CoefficientList[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/(x n), x],
   {n, 1, 10}] // Flatten (* Robert A. Russell, Feb 23 2018 *)
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]]
Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/n,{n,1,12},{k,1,n}] // Flatten (* Robert A. Russell, Oct 16 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(CyclicPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = (1/n)*Sum_{d|n} phi(d)*A(d,n/d,k), where 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)). - Robert A. Russell, Oct 16 2018

A056361 Number of bracelet structures using exactly six different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 27, 171, 1249, 8389, 56079, 360430, 2272601, 14037552, 85516454, 514976658, 3074986408, 18239677629, 107654219304, 632996894928, 3711499493421, 21716765203045, 126880009607690, 740528525043982, 4319138789721875, 25181507049874027, 146788327084831744

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 6 of A152176.

Cf. A056299, A056355, A056356.

a(n) = A056356(n) - A056355(n).

Terms a(25) and beyond from Andrew Howroyd, Oct 24 2019

A320646 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 6 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 9, 125, 1054, 7928, 54383, 356594, 2259504, 14008733, 85422360, 514773336, 3074341497, 18238301412, 107649939612, 632987843336, 3711471738408, 21716706883190, 126879832615600, 740528154956264, 4319137675225128, 25181504728152534, 146788320134425736, 855660631677225738, 4988501691655508510, 29089896998939710698

Offset: 1

Robert A. Russell, Oct 19 2018

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 A056299 and A304976, which can be used in conjunction with the first formula.

			For a(8)=9, the chiral pairs are AABACDEF-AABCDEAF, AABCADEF-AABCDAEF, AABCBDEF-AABCDEFE, AABCDBEF-AABCDEFD, AABCDEBF-AABCDEFC, AABCDCEF-AABCDEDF, ABACDEBF-ABACDEBF, ABCADBEF-ABCADECF, and ABCDAEBF-ABCADBEF.

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.

Column 6 of A320647.

Cf. A056299 (oriented), A056361 (unoriented), A304976 (achiral).

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

a(n) = (A056299(n) - A304976(n)) / 2 = A056299(n) - A056361(n) = A056361(n) - A304976(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 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)).

A056307 Number of primitive (period n) n-bead necklace structures using exactly six different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 36, 296, 2303, 16317, 110461, 717024, 4532102, 28046285, 170938778, 1029749994, 6149327608, 36477979041, 215304156613, 1265984738261, 7422971215512, 43433472086235, 253759842112792

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 6 of A107424.

Cf. A056291.

Sum mu(d)*A056299(n/d) where d|n. Alternatively, A056302(n)-A056301(n)..

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A152175 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations.

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A056361 Number of bracelet structures using exactly six different colored beads.

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A320646 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 6 colors (subsets).

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A056307 Number of primitive (period n) n-bead necklace structures using exactly six different colored beads.

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