A152176 -id:A152176 - 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

A284949 Triangle read by rows: T(n,k) = number of reversible string structures of length n using exactly k different symbols.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 9, 15, 6, 1, 1, 19, 50, 37, 9, 1, 1, 35, 160, 183, 76, 12, 1, 1, 71, 502, 877, 542, 142, 16, 1, 1, 135, 1545, 3930, 3523, 1346, 242, 20, 1, 1, 271, 4730, 17185, 21393, 11511, 2980, 390, 25, 1

Offset: 1

Andrew Howroyd, Apr 06 2017

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.
Number of k-block partitions of an n-set up to reflection.
T(n,k) = pi_k(P_n) which is the number of non-equivalent partitions of the path on n vertices, with exactly k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

			Triangle begins:
1;
1,   1;
1,   2,    1;
1,   5,    4,     1;
1,   9,   15,     6,     1;
1,  19,   50,    37,     9,     1;
1,  35,  160,   183,    76,    12,    1;
1,  71,  502,   877,   542,   142,   16,   1;
1, 135, 1545,  3930,  3523,  1346,  242,  20,  1;
1, 271, 4730, 17185, 21393, 11511, 2980, 390, 25, 1;

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
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
Mohammad Hadi Shekarriz, GAP Program

Columns 2..6 are A056326, A056327, A056328, A056329, A056330.
Row sums are A103293.
Partial row sums include A005418, A001998(n-1), A056323, A056324, A056325.
Cf. A277504, A008277 (set partitions), A152175 (up to rotation), A152176 (up to rotation and reflection), A304972 (achiral patterns).

(* achiral color patterns for row of n colors containing k different colors *)
Ach[n_, k_] := Ach[n, k] = Switch[k, 0, If[0==n, 1, 0], 1, If[n>0, 1, 0],
   (* else *) _, If[OddQ[n],
   Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],
   Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1] + 2^i Ach[n-2-2i, k-2]),
   {i, 0, n/2-1}]]]
Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 15}, {k, 1, n}] // Flatten
(* Robert A. Russell, Feb 10 2018 *)

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

\\ Ach is A304972 as square matrix.
Ach(n)={my(M=matrix(n,n,i,k,i>=k)); for(i=3, n, for(k=2, n, M[i,k]=k*M[i-2,k] + M[i-2,k-1] + if(k>2, M[i-2,k-2]))); M}
T(n)={(matrix(n, n, i, k, stirling(i, k, 2)) + Ach(n))/2}
{ my(A=T(10)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Sep 18 2019

A276543 Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 13, 11, 3, 1, 0, 8, 31, 33, 16, 3, 1, 0, 14, 80, 136, 85, 27, 4, 1, 0, 21, 201, 478, 434, 171, 37, 4, 1, 0, 39, 533, 1849, 2270, 1249, 338, 54, 5, 1, 0, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1

Offset: 1

Andrew Howroyd, Apr 09 2017

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

			Triangle starts:
  1
  0  1
  0  1   1
  0  2   2    1
  0  3   5    2    1
  0  5  13   11    3    1
  0  8  31   33   16    3   1
  0 14  80  136   85   27   4  1
  0 21 201  478  434  171  37  4 1
  0 39 533 1849 2270 1249 338 54 5 1
  ...

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

Columns 1-6 are A063524, A056366, A056367, A056368, A056369, A056370.
Partial row sums include A000046, A056362, A056363, A056364, A056365.
Row sums are A276548.
Cf. A276550, A152175, A152176, A107424, A137651, A276544, A304972.

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
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))]))}
T(n)={my(M=(R(n)+Ach(n))/2); Mat(vectorv(n,n,sumdiv(n, d, moebius(d)*M[n/d,])))}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).

A056353 Number of bracelet structures using a maximum of three different colored beads.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 22, 40, 100, 225, 582, 1464, 3960, 10585, 29252, 80819, 226530, 636321, 1800562, 5107480, 14548946, 41538916, 118929384, 341187048, 980842804, 2824561089, 8147557742, 23536592235, 68087343148, 197216119545, 571924754778, 1660419530056, 4825588205920

Offset: 0

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 = 0..200
R. M. Thompson and R. T. Downs, Systematic generation of all nonequivalent closest packed stacking sequences of length N using group theory, Acta Cryst. B57 (2001), 766-771; B58 (2002), 153.

Cf. A002076, A000011, A027671, A114438, A152176.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..3} A152176(n, k) for n > 0. - Andrew Howroyd, Oct 25 2019

a(0)=1 prepended and terms a(28) and beyond from Andrew Howroyd, Oct 25 2019

A309784 T(n,k) is the number of non-equivalent distinguishing coloring partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 2, 1, 0, 1, 8, 10, 3, 1, 0, 1, 25, 32, 16, 3, 1, 0, 4, 62, 129, 84, 27, 4, 1, 0, 7, 176, 468, 433, 171, 37, 4, 1, 0, 18, 470, 1806, 2260, 1248, 338, 54, 5, 1, 0, 31, 1311, 6780, 11515, 8388, 3056, 590, 70, 5, 1, 0, 70, 3620, 25917, 58312, 56065, 26695, 6907, 1014, 96, 6, 1

Offset: 1

Mohammad Hadi Shekarriz, Aug 17 2019

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. A distinguishing coloring partition of a graph G is a partition of the vertices of G such that it induces a distinguishing coloring for G. We say two distinguishing coloring partitions P1 and P2 of G are equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. Given a graph G, we use the notation psi_k(G) to denote the number of non-equivalent distinguishing coloring partitions of G with exactly k parts. For n>=3, this sequence gives T(n,k) = psi_k(C_n), i.e., the number of non-equivalent distinguishing coloring partitions of the cycle C_n on n vertices with exactly k parts.
T(n,k) is the number of primitive (period n) n-bead bracelet structures which are not periodic palindromes using exactly k different colored beads. - Andrew Howroyd, Sep 20 2019

			The triangle begins:
  0;
  0,  0;
  0,  0,   1;
  0,  0,   1,    1;
  0,  0,   4,    2,    1;
  0,  1,   8,   10,    3,    1;
  0,  1,  25,   32,   16,    3,   1;
  0,  4,  62,  129,   84,   27,   4,  1;
  0,  7, 176,  468,  433,  171,  37,  4, 1;
  0, 18, 470, 1806, 2260, 1248, 338, 54, 5, 1;
  ...
For n=6, we can partition the vertices of C_6 into exactly 3 parts in 8 ways such that all these partitions induce distinguishing colorings for C_6 and that all the 8 partitions are non-equivalent. The partitions are as follows:
    { { 1 }, { 2 }, { 3, 4, 5, 6 } }
    { { 1 }, { 2, 3 }, { 4, 5, 6 } }
    { { 1 }, { 2, 3, 4, 6 }, { 5 } }
    { { 1 }, { 2, 3, 5 }, { 4, 6 } }
    { { 1 }, { 2, 3, 6 }, { 4, 5 } }
    { { 1 }, { 2, 4, 5 }, { 3, 6 } }
    { { 1, 2 }, { 3, 4 }, { 5, 6 } }
    { { 1, 2 }, { 3, 5 }, { 4, 6 } }
For n=6, the above 8 partitions can be written as the following 3 colored bracelet structures: ABCCCC, ABBCCC, ABBBCB, ABBCBC, ABBCCB, ABCBBC, AABBCC, AABCBC. - _Andrew Howroyd_, Sep 22 2019

Andrew Howroyd, Table of n, a(n) for n = 1..1275
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
Mohammad Hadi Shekarriz, GAP Program

Column k=2 appears to be A011948.
Columns k=3..4 are A328038, A328039.
Row sums are A328035.
Cf. A107424, A152175, A152176, A276543, A276544, A285012, A285037, A309748, A309785.

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
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))]))}
T(n)={my(A=Ach(n), M=R(n), S=matrix(n, n, n, k, stirling(n, k, 2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(M[n/d,] + A[n/d,])/2 - moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Oct 02 2019

T(n,k) = A276543(n,k) - A285037(n,k). - Andrew Howroyd, Sep 20 2019

T(10,6) corrected by Mohammad Hadi Shekarriz, Sep 28 2019

a(56)-a(78) from Andrew Howroyd, Sep 28 2019

A320647 Triangle read by rows: T(n,k) is the number of chiral pairs of cycles of length n (1) with a color pattern of exactly k colors or equivalently (2) partitioned into k nonempty subsets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 1, 12, 17, 4, 0, 0, 0, 2, 44, 84, 51, 9, 0, 0, 0, 7, 137, 388, 339, 125, 15, 0, 0, 0, 12, 408, 1586, 2010, 1054, 258, 24, 0, 0, 0, 31, 1190, 6405, 10900, 7928, 2761, 490, 35, 0, 0, 0, 58, 3416, 24927, 56700, 54383, 25680, 6392, 859, 51, 0, 0, 0, 126, 9730, 96404, 286888, 356594, 218246, 72284, 13472, 1420, 68, 0, 0

Offset: 1

Robert A. Russell, Oct 18 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.

			The triangle begins with T(1,1):
  0;
  0,   0;
  0,   0,    0;
  0,   0,    0,     0;
  0,   0,    0,     0,      0;
  0,   0,    4,     2,      0,      0;
  0,   1,   12,    17,      4,      0,      0;
  0,   2,   44,    84,     51,      9,      0,     0;
  0,   7,  137,   388,    339,    125,     15,     0,     0;
  0,  12,  408,  1586,   2010,   1054,    258,    24,     0,    0;
  0,  31, 1190,  6405,  10900,   7928,   2761,   490,    35,    0,  0;
  0,  58, 3416, 24927,  56700,  54383,  25680,  6392,   859,   51,  0, 0;
  0, 126, 9730, 96404, 286888, 356594, 218246, 72284, 13472, 1420, 68, 0, 0;
  ...
For T(6,3)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
For T(6,4)=2, the chiral pairs are AABACD-AABCAD and AABCBD-AABCDC.

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.

Columns 1-6 are A000004, A059053, A320643, A320644, A320645, A320646.
Row sums are A320749.
Cf. A152175 (oriented), A152176 (unoriented), A304972 (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]]
Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n)-Ach[n,k]/2,{n,12},{k,n}] // Flatten

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
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))]))}
T(n)={(R(n) - Ach(n))/2}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = (A152175(n,k) - A304972(n,k)) / 2 = A152175(n,k) - A152176(n,k) = A152176(n,k) - A304972(n,k).

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

A213940 Triangle with entry a(n,m) giving the number of bracelets of n beads (dihedral D_n symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 3, 6, 6, 12, 1, 7, 20, 26, 30, 60, 1, 8, 40, 93, 150, 180, 360, 1, 18, 106, 424, 633, 1050, 1260, 2520, 1, 22, 304, 1180, 3260, 5040, 8400, 10080, 20160, 1, 46, 731, 4844, 16212, 29244, 45360, 75600, 90720, 181440

Offset: 1

Wolfdieter Lang, Jul 20 2012

This triangle is obtained from the partition array A213939 by summing in row n, for n>=1, all entries related to partitions of n with the same number of parts m.
a(n,m) is the number of bracelets of n beads (dihedral D_n symmetry) corresponding to the representative color multinomials obtained from all partitions of n with m parts by 'exponentiation', hence only m from the available n colors are present. As a representative multinomial of each of the p(n,m)=A008284(n,m) such m-color classes we take the one where the considered m part partition of n, [p[1],...,p[m]], written in nonincreasing order, is distributed as exponents on the color indices like c[1]^p[1]*...*c[m]^p[m]. That is only the first m colors from the n available ones are involved.
See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.
The row sums of this triangle coincide with the ones of array A213939, and they are given by A213943.
Number of n-length bracelets w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226874). - Andrew Howroyd, Sep 26 2017

			n\m  1  2   3    4     5     6     7     8     9     10 ...
1    1
2    1  1
3    1  1   1
4    1  3   2    3
5    1  3   6    6    12
6    1  7  20   26    30    60
7    1  8  40   93   150   180   360
8    1 18 106  424   633  1050  1260  2520
9    1 22 304 1180  3260  5040  8400 10080 20160
10   1 46 731 4844 16212 29244 45360 75600 90720 181440
...
a(5,3) = 2 + 4 = 6, from A213939(5,4) + A213939(5,5), because k(5,3,1) = 4 and p(5,3) = 2.
a(2,1) = 1 because the partition [2] of n=2 with part number m=1 corresponds to the representative color multinomial (here monomial) c[1]^2 = c[1]*c[1], and there is one such representative bracelet. There is another bracelet color monomial in this class of n=2 colors where only m=1 color is active: c[2]*c[2]. See the triangle entry A213941(2,1)=2. The same holds for the necklace case.
a(3,1) = 1 from the color monomial representative c[1]^3. This class has 2 other members: c[2]^3 and c[3]^3. See A213941(3,1)=3. The same holds for the necklace case.
Like in the necklace case one has in general a(n,1)=1 and A213941(n,1) = n from the partition [n] providing the color signature and a representative c[1]^n.
a(3,2) = 1 from the representative color multinomial c[1]^2*c[2] (from the m=2 partition [2,1] of n=3) leading to just one representative bracelet (and necklace) cyclic(112) (when one uses j for color c[j]). The whole class consists of A213941(3,2)=6 bracelets (or necklaces): cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332).
a(3,3) = 1. The representative color multinomial is c[1]*c[2]*c[3] (from the m=3 partition [1,1,1]). There is only one bracelet cyclic(1,2,3) which constitutes already the whole class (A213941(3,3)=1). The necklace cyclic(1,3,2) becomes equivalent under D_3.
a(4,2) = 3 from two representative color multinomials c[1]^3*c[2] and c[1]^2*c[2]^2 (from the two m=2 partitions of n=4: [3,1] and [2,2]). The first one has one representative bracelet, namely cyclic(1112), the second one leads to the two representative bracelets: cyclic(1122) and cyclic(1212). Together these are the 3 bracelets counted by a(4,2). The first color class c[.]^3*c[.] consists of 4*3=12 bracelets, when all 4 colors are used. The second one consists of 2*6=12 bracelets. Together they sum up to the 24 bracelets counted by A213941(4,2). In this example the necklace case does not differ from the bracelet one.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=2..5 are A213942, A214307, A214309, A214311.
Cf. A008284, A213939, A213941, A213943 (row sums), A226874, A152176.
Cf. A213934 (cyclic symmetry).

Cyc(v)={my(s=vecsum(v)); sumdiv(gcd(v), d, eulerphi(d)*(s/d)!/prod(i=1, #v, (v[i]/d)!))/s}
CPal(v)={my(odds=#select(t->t%2,v), s=vecsum(v));  if(odds>2, 0, ((s-odds)/2)!/prod(i=1, #v, (v[i]\2)!))}
T(n,k)={my(t=0); forpart(p=n, t+=Cyc(Vec(p))+CPal(Vec(p)), [1,n], [k,k]); t/2}
for(n=1, 10, for(k=1,n, print1(T(n,k), ", ")); print); \\ Andrew Howroyd, Sep 26 2017

\\ faster version; here U is A226874 as vector of polynomials.
U(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
T(n)={my(t=U(n)); vector(n, n, vector(n, k, ((1/n)*sumdiv(n, d, eulerphi(n/d) * polcoeff(t[d+1], k)) + if(n%2, sum(d=0, (n-1)/2, binomial((n-1)/2, d)*polcoeff(t[d+1], (k-1))), polcoeff(t[n/2+1], k) + sum(d=0, n/2-1, binomial(n/2-1, d)*(2^d + if(d%2, 0, binomial(d, d/2)))*polcoeff(t[n/2-d], k-2))/2))/2))}
{ my(t=T(10)); for(n=1, #t, print(t[n])) } \\ Andrew Howroyd, Dec 22 2017

a(n,m) = Sum_{j=1..p(n,m)}A213939(n,k(n,m,1)+j-1), with k(n,m,1) the position where in the list of partitions of n in A-St order the first with m parts appears, and p(n,m) is the number of partitions of n with m parts shown in the array A008284. E.g., n=5, m=3: k(5,3,1)=4, p(5,3)=2.

A056354 Number of bracelet structures using a maximum of four different colored beads.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 33, 73, 237, 703, 2433, 8309, 30108, 108991, 403262, 1497070, 5607437, 21076571, 79595990, 301492045, 1145560579, 4363503684, 16660204452, 63741248201, 244339646708, 938255682551, 3608668388957, 13900021844558, 53614340398327, 207062143625711

Offset: 0

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 = 0..200

Cf. A056292, A152176.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..4} A152176(n, k) for n > 0. - Andrew Howroyd, Oct 25 2019

a(0)=1 prepended and terms a(26) and beyond from Andrew Howroyd, Oct 25 2019

A056355 Number of bracelet structures using a maximum of five different colored beads.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 36, 89, 322, 1137, 4704, 19839, 88508, 399680, 1839947, 8533488, 39893901, 187393550, 884153396, 4185740195, 19876594537, 94633345608, 451615319433, 2159769331317, 10348546548695, 49672000435724, 238804871206358, 1149792978954373, 5543621482141513

Offset: 0

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 = 0..200

Cf. A056293, A152176.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..5} A152176(n, k) for n > 0. - Andrew Howroyd, Oct 25 2019

a(0)=1 prepended and terms a(25) and beyond from Andrew Howroyd, Oct 25 2019

A084708 Number of set partitions up to rotations and reflections.

Original entry on oeis.org

1, 2, 3, 7, 12, 37, 93, 354, 1350, 6351, 31950, 179307, 1071265, 6845581, 46162583, 327731950, 2437753740, 18948599220, 153498350745, 1293123243928, 11306475314467, 102425554299516, 959826755336242, 9290811905391501

Offset: 1

Wouter Meeussen, Jul 02 2003

nonn

Combines the symmetry operations of A080107 and A084423.

Equivalently, number of n-bead bracelets using any number of unlabeled (interchangable) colors. - Andrew Howroyd, Sep 25 2017

			SetPartitions[6] is the first to decompose differently from A084423: 4 cycles of length 1, 2 of 2, 9 of 3, 16 of 6, 6 of 12.
a(7) = 1 + A056357(7) + A056358(7) + A056359(7) + A056360(7) + A056361(7) + 1 = 1 + 8 + 31 + 33 + 16 + 3 + 1 = 93.

Michael De Vlieger, Table of n, a(n) for n = 1..577
Colin Adams, Chaim Even-Zohar, Jonah Greenberg, Reuben Kaufman, David Lee, Darin Li, Dustin Ping, Theodore Sandstrom, and Xiwen Wang, Virtual Multicrossings and Petal Diagrams for Virtual Knots and Links, arXiv:2103.08314 [math.GT], 2021.
Tilman Piesk, Partition related number triangles
N. J. A. Sloane, Generating functions [From _Wouter Meeussen_, Dec 06 2008]

Cf. A080107, A084423, A080510, A002872, A002874, A141003, A036075, A141004, A036077, A152176.

<A080107 *); Table[{Length[ # ], First[ # ]}&/@ Split[Sort[Length/@Split[Sort[First[Sort[Flatten[ {#, Map[Sort, (#/. i_Integer:>w+1-i), 2]}& @(NestList[Sort[Sort/@(#/. i_Integer :> Mod[i+1, w, 1])]&, #, w]), 1]]]&/@SetPartitions[w]]]]], {w, 1, 10}]
u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; a[n_]:=1/n*Plus@@(EulerPhi[ # ]u[Quotient[n,# ],# ]&/@Divisors[n]); Table[a[n]/2+If[EvenQ[n],u[n/2,2],Sum[Binomial[n/2-1/2,k] u[k,2], {k,0,n/2-1/2}]]/2,{n,40}] (* Wouter Meeussen, Dec 06 2008 *)

a(n) = (A080107(n)+A084423(n))/2. - Wouter Meeussen and Vladeta Jovovic, Nov 28 2008

a(12) from Vladeta Jovovic, Jul 15 2007

More terms from Vladeta Jovovic's formula given in Mathematica line. - Wouter Meeussen, Dec 06 2008

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.

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A284949 Triangle read by rows: T(n,k) = number of reversible string structures of length n using exactly k different symbols.

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A276543 Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.

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A056353 Number of bracelet structures using a maximum of three different colored beads.

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A309784 T(n,k) is the number of non-equivalent distinguishing coloring partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

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A320647 Triangle read by rows: T(n,k) is the number of chiral pairs of cycles of length n (1) with a color pattern of exactly k colors or equivalently (2) partitioned into k nonempty subsets.

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A213940 Triangle with entry a(n,m) giving the number of bracelets of n beads (dihedral D_n symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

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