A152175 -id:A152175 - cp's OEIS frontend

A152176 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations and reflections.

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 14, 11, 3, 1, 1, 8, 31, 33, 16, 3, 1, 1, 17, 82, 137, 85, 27, 4, 1, 1, 22, 202, 478, 434, 171, 37, 4, 1, 1, 43, 538, 1851, 2271, 1249, 338, 54, 5, 1, 1, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1, 1, 121, 3838, 26148

Offset: 1

Vladeta Jovovic, Nov 27 2008

Number of bracelet structures of length n using exactly k different colored beads. Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure. - Andrew Howroyd, Apr 06 2017
The number of achiral structures (A) is given in A140735 (odd n) and A293181 (even n).  The number of achiral structures plus twice the number of chiral pairs (A+2C) is given in A152175.  These can be used to determine A+C by taking half their average, as is done in the Mathematica program. - Robert A. Russell, Feb 24 2018
T(n,k)=pi_k(C_n) which is the number of non-equivalent partitions of the cycle 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,  1,   1;
  1,  3,   2,    1;
  1,  3,   5,    2,    1;
  1,  7,  14,   11,    3,    1;
  1,  8,  31,   33,   16,    3,   1;
  1, 17,  82,  137,   85,   27,   4,  1;
  1, 22, 202,  478,  434,  171,  37,  4, 1;
  1, 43, 538, 1851, 2271, 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
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Tilman Piesk, Partition related number triangles
Marko Riedel, Bracelets with swappable colors classified by the distribution of colors, Power Group Enumeration algorithm
Marko Riedel, Maple code for the number of bracelets with some number of swappable colors by Power Group Enumeration
Mohammad Hadi Shekarriz, GAP Program

Columns 2-6 are A056357, A056358, A056359, A056360, A056361.
Row sums are A084708.
Partial row sums include A000011, A056353, A056354, A056355, A056356.
Cf. A081720, A273891, A008277 (set partitions), A284949 (up to reflection), A152175 (up to rotation).

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]];
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}]]] (* achiral loops of length n, k colors *)
Table[(CoefficientList[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/(x n), x]
+ Table[Ach[n, k],{k,1,n}])/2, {n, 1, 20}] // Flatten (* Robert A. Russell, Feb 24 2018 *)

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

\\ 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

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).

A084423 Set partitions up to rotations.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 43, 127, 544, 2361, 11703, 61690, 351773, 2126497, 13639372, 92197523, 655035769, 4874404108, 37893370473, 306986431847, 2586209749712, 22612848403571, 204850732480285, 1919652428481930, 18581619724363401, 185543613289200949

Offset: 0

Wouter Meeussen, Jun 26 2003

Partitions of n objects distinct under the cyclic group, C_n. By comparison the partition numbers (A000041) are the partitions distinct under the symmetric group, S_n and the set partitions are those distinct under the discrete group containing only the identity. - Franklin T. Adams-Watters, Jun 09 2008

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

			Of the Bell(4) = 15 set partitions of 4, only 7 remain distinct under rotation:
  {{1,2,3,4}},
  {{1}, {2,3,4}},
  {{1,2}, {3,4}},
  {{1,3}, {2,4}},
  {{1}, {2}, {3,4}},
  {{1}, {3}, {2,4}},
  {{1}, {2}, {3}, {4}}.

Alois P. Heinz, Table of n, a(n) for n = 0..500 (first 61 terms from Franklin T. Adams-Watters)
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.
Robert M. Dickau, Bell number diagrams
Wouter Meeussen, Set Partitions Up To Rotation
Tilman Piesk, Partition related number triangles

Cf. A080107, A084708, A152175.

<Mod[i+1, n, 1])]&, #, n]]]& /@ SetPartitions[n]]; Table[ Length[ shrink[k]], {k, 11}]
(* Second program (not needing Combinatorica): *)
u[0, ] = 1; u[k, j_] := u[k, j] = Sum[Binomial[k-1, i-1]*Sum[u[k-i, j]*d^(i-1), {d, Divisors[j]}], {i, 1, k}]; a[n_] := Sum[EulerPhi[j]*u[n/j, j], {j, Divisors[n]}]/n; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 14 2012, after Franklin T. Adams-Watters *)

U(k, j) = if(k==0,1,sum(i=1,k,binomial(k-1,i-1)*sumdiv(j,d,U(k-i,j) *d^(i-1)))) /* U is unoptimized; should remember previous values. */
a(n) = sumdiv(n,j,eulerphi(j)*U(n\j,j))/n \\ Franklin T. Adams-Watters, Jun 09 2008

seq(n)={Vec(1 + intformal(sum(m=1, n, eulerphi(m)*subst(serlaplace(-1 + exp(sumdiv(m, d, (exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))} \\ Andrew Howroyd, Sep 20 2019

a(p) = (Bell(p)+2*(p-1))/p for prime p; cf. A079609. - Vladeta Jovovic, Jul 04 2003
U(k,j) = 1 if k=0, else Sum_{i=1..k} C(k-1,i-1) Sum_{d|j} U(k-i,j)*d^{i-1}. Then a(n) = (Sum_{j|n} phi(j)*U(n/j,j))/n. (U(k,j) is the number of partitions invariant under a permutation with k cycles of j objects each.) - Franklin T. Adams-Watters, Jun 09 2008
a(n) = [n==0] + [n>0] * (1/n) * Sum_{d|n} phi(d) * A162663(n/d,d). - Robert A. Russell, Jun 10 2018
From Richard L. Ollerton, May 09 2021: (Start)
For n >= 1:
a(n) = (1/n)*Sum_{k=1..n} A162663(gcd(n,k),n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} A162663(n/gcd(n,k),gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

More terms from Robert G. Wilson v, Jun 27 2003

More terms from Franklin T. Adams-Watters, Jun 09 2008

A107424 Triangle read by rows: T(n, k) is the number of primitive (period n) n-bead necklace structures with k different colors. Only includes structures that contain all k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 17, 13, 3, 1, 0, 9, 43, 50, 20, 3, 1, 0, 16, 124, 220, 136, 36, 4, 1, 0, 28, 338, 866, 773, 296, 52, 4, 1, 0, 51, 941, 3435, 4280, 2303, 596, 78, 5, 1, 0, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 0, 170, 7234, 51061

Offset: 1

David Wasserman, May 26 2005

This classification is concerned with which beads are the same color, not with the colors themselves, so bbabcd is the same structure as aabacd. Cyclic permutations are also the same structure, e.g. abacda is also the same structure. However, order matters: the reverse of aabacd is equivalent to aabcad, which is also on the list.

			T(6, 4) = 13: {aaabcd, aabacd, aabcad, abacad, aabbcd, aabcbd, aabcdb, aacbbd, aacbdb, ababcd, abacbd, acabdb, abcabd}.
From _Andrew Howroyd_, Apr 09 2017 (Start)
Triangle starts:
1
0  1
0  1   1
0  2   2    1
0  3   5    2    1
0  5  17   13    3    1
0  9  43   50   20    3   1
0 16 124  220  136   36   4  1
0 28 338  866  773  296  52  4 1
0 51 941 3435 4280 2303 596 78 5 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 2-6 are A056303, A056304, A056305, A056306, A056307.
Partial row sums include A000048, A002075, A056300, A056301, A056302.
Row sums are A276547.
Cf. A074650, A152175, A276543, A137651, A276544.

A[d_, n_] := A[d, n] = Which[n == 0, 1, n == 1, DivisorSum[d, x^# &], d == 1, Sum[StirlingS2[n, k] x^k, {k, 0, n}], True, Expand[A[d, 1] A[d, n-1] + D[A[d, n-1], x] x]];
B[n_, k_] := Coefficient[DivisorSum[n, EulerPhi[#] A[#, n/#]&]/n/x, x, k];
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] B[#, k]&];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 06 2018, after Andrew Howroyd and Robert A. Russell *)

\\ here R(n) is A152175 as square matrix.
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)); matrix(n, n, i, k, sumdiv(i, d, moebius(i/d)*M[d,k]))}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 09 2020

T(n, k) = Sum_{d|n} mu(n/d) * A152175(d, k). - Andrew Howroyd, Apr 09 2017

A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.

Original entry on oeis.org

0, 1, 4, 1, 7, 31, 3, 23, 179, 2107, 3, 55, 1095, 26271, 671103, 7, 189, 7327, 350063, 17896831, 954459519, 9, 595, 49939, 4794087, 490853415, 52357746895, 5744387279871, 19, 2101, 349715, 67115111, 13743921631, 2932032057731, 643371380132743, 144115188277194943, 29, 7315, 2485591, 954444607, 390937468407, 166799988703927, 73201365371896619

Offset: 1

Marko Riedel, Nov 08 2017

Two colorings are equivalent if there is a permutation of the colors that takes one to the other in addition to translational symmetries on the torus. (Power Group Enumeration.)

			For the 2 X 2 grid and two colors we find T(2,2) = 4:
  +---+  +---+  +---+  +---+
  |X| |  |X| |  |X|X|  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+
  | | |  | |X|  | | |  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequences A294791, A294792, A294793, A294794.

Cf. A152175, A294684, A294685, A294686, A294687, A294792, A294793, A294794, A295197. T(n,1) is A056295.

T(n,k) = (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma) with Q=2. The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.

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)).

A056295 Number of n-bead necklace structures using exactly two different colored beads.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 9, 19, 29, 55, 93, 179, 315, 595, 1095, 2067, 3855, 7315, 13797, 26271, 49939, 95419, 182361, 349715, 671091, 1290871, 2485533, 4794087, 9256395, 17896831, 34636833, 67110931, 130150587, 252648991, 490853415, 954444607, 1857283155, 3616828363

Offset: 1

Marks R. Nester

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

			For a(7) = 9, the color patterns are AAAAAAB, AAAAABB, AAAABAB, AAAABBB, AAABAAB, AABAABB, AABABAB, AAABABB, and AAABBAB. The first seven are achiral; the last two are a chiral pair. - _Robert A. Russell_, Mar 08 2018

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.]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.

Column 2 of A152175.

Cf. A000013, A052823.

Maple
```
See A000013.
```

Table[DivisorSum[n, EulerPhi[#] If[OddQ[#], StirlingS2[n/#, 2], StirlingS2[n/#+1, 2]]&]/n, {n,1,30}] (* Robert A. Russell, Feb 20 2018 *)

a(n) = A000013(n) - 1.
From Robert A. Russell, Mar 08 2018: (Start)
G.f.: Sum_{ d>0 } phi(d)*(2*log(1-x^d) - (1+[d == 0 mod 2])*log(1-2*x^d)) / (2*d);
a(n) = (1/n)*Sum_{d|n} phi(d) * S2(n/d + [d == 0 mod 2], 2), where S2(n, k) is the Stirling subset number, A008277. (End)

A320742 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a cycle of length n using k or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 6, 13, 2, 0, 0, 0, 0, 0, 0, 6, 30, 46, 7, 0, 0, 0, 0, 0, 0, 6, 34, 130, 144, 12, 0, 0, 0, 0, 0, 0, 6, 34, 181, 532, 420, 31, 0, 0, 0, 0, 0, 0, 6, 34, 190, 871, 2006, 1221, 58, 0, 0, 0, 0, 0, 0, 6, 34, 190, 996, 4016, 7626, 3474, 126, 0, 0, 0, 0, 0, 0, 6, 34, 190, 1011, 5070, 18526, 28401, 9856, 234, 0

Offset: 1

Robert A. Russell, Oct 21 2018

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.

			Array begins with T(1,1):
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    4     6     6      6      6      6      6      6      6      6 ...
0  1   13    30    34     34     34     34     34     34     34     34 ...
0  2   46   130   181    190    190    190    190    190    190    190 ...
0  7  144   532   871    996   1011   1011   1011   1011   1011   1011 ...
0 12  420  2006  4016   5070   5328   5352   5352   5352   5352   5352 ...
0 31 1221  7626 18526  26454  29215  29705  29740  29740  29740  29740 ...
0 58 3474 28401 85101 139484 165164 171556 172415 172466 172466 172466 ...
For T(6,4)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, 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.

Partial row sums of A320647.
Columns 1-6 are A000004, A059053, A320743, A320744, A320745, A320746
For increasing k, columns converge to A320749.
Cf. A320747 (oriented), A320748 (unoriented), A305749 (achiral).

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]] (* A304972 *)
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j,k-n+1}], {k,15}, {n,k}] // 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)={my(M=(R(n) - Ach(n))/2); for(i=2, n, M[,i] += M[,i-1]); M}
{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019

T(n,k) = Sum_{j=1..k} -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)).

T(n,k) = (A320747(n,k) - A305749(n,k)) / 2 = A320747(n,k) - A320748(n,k)= A320748(n,k) - A305749(n,k).

cp's OEIS Frontend

A152176 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations and reflections.

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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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A084423 Set partitions up to rotations.

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A107424 Triangle read by rows: T(n, k) is the number of primitive (period n) n-bead necklace structures with k different colors. Only includes structures that contain all k colors.

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A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.

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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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