A087854 -id:A087854 - cp's OEIS frontend

A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, 147512053, 2340964372, 40527565261, 760095929840, 15352212731933, 332228417657960, 7668868648772701, 188085259070219000, 4884294069438337429

Offset: 1

Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

Original name: a(n) = number of necklaces of n beads with up to n unlabeled colors.

The Moebius transform of this sequence is A060223.

			a(3) = 5 since there are the following length 3 words up to rotation:
     111,  112, 122, 123, 132.
a(4) = 20 since there are the following length 4 words up to rotation:
     1111,
     1112, 1122, 1212, 1222,
     1123, 1132, 1213, 1223, 1232, 1233, 1322, 1323, 1332,
     1234, 1243, 1324, 1342, 1423, 1432.

G. C. Greubel, Table of n, a(n) for n = 1..420
M. Goebel, On the number of special permutation-invariant orbits and terms, in: Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.); see p. 509 (stated as an open problem).
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Eric Weisstein's world of Mathematics, Necklaces.

Cf. A000670, A008277, A019537, A060223.

Row sums of A087854. - Philippe Deléham

Needs["DiscreteMath`Combinatorica`"];
mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]];
neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li];Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n];
Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
(* second program: *)
a[n_] := Sum[DivisorSum[n, EulerPhi[#]*StirlingS2[n/#, k] k! &]/n, {k, 1, n}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 31 2016, after Philippe Deléham *)

a(n) = sum(k=1, n, sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)*k!)/n); \\ Michel Marcus, Mar 31 2016

See Mathematica code.
a(n) ~ (n-1)! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Jul 21 2019
From Petros Hadjicostas, Aug 19 2019: (Start)
The first formula is due to Philippe Deléham from the Crossrefs (see also the programs below). The second one follows easily from the first one. The third one follows from the second one using the associative property of Dirichlet convolutions.
a(n) = Sum_{k = 1..n} (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n, k) = Stirling numbers of 2nd kind (A008277).
a(n) = (1/n) * Sum_{d|n} phi(d) * A000670(n/d).
a(n) = Sum_{d|n} A060223(d).
(End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = (1/n)*Sum_{k=1..n} A000670(gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} A000670(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Edited by Wouter Meeussen, Aug 06 2002
Corrected by T. D. Noe, Oct 31 2006
Edited by Andrew Howroyd, Aug 19 2019

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

Original entry on oeis.org

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

A254040 Number T(n,k) of primitive (= aperiodic) n-bead necklaces with colored beads of exactly k different colors; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 9, 6, 0, 0, 6, 30, 48, 24, 0, 0, 9, 89, 260, 300, 120, 0, 0, 18, 258, 1200, 2400, 2160, 720, 0, 0, 30, 720, 5100, 15750, 23940, 17640, 5040, 0, 0, 56, 2016, 20720, 92680, 211680, 258720, 161280, 40320

Offset: 0

Alois P. Heinz, Jan 23 2015

Turning over the necklaces is not allowed.

With other words: T(n,k) is the number of normal Lyndon words of length n and maximum k, where a finite sequence is normal if it spans an initial interval of positive integers. - Gus Wiseman, Dec 22 2017

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0,  1;
  0, 0,  2,   2;
  0, 0,  3,   9,    6;
  0, 0,  6,  30,   48,    24;
  0, 0,  9,  89,  260,   300,   120;
  0, 0, 18, 258, 1200,  2400,  2160,   720;
  0, 0, 30, 720, 5100, 15750, 23940, 17640, 5040;
  ...
The T(4,3) = 9 normal Lyndon words of length 4 with maximum 3 are: 1233, 1323, 1332, 1223, 1232, 1322, 1123, 1132, 1213. - _Gus Wiseman_, Dec 22 2017

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A063524, A001037 (for n>1), A056288, A056289, A056290, A056291, A254079, A254080, A254081, A254082.
Row sums give A060223.
Main diagonal and lower diagonal give: A000142, A074143.
T(2n,n) gives A254083.
Cf. A074650, A087854.
Cf. A000670, A000740, A008683, A019536, A059966, A185700, A296372, A296373, A296657.

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(mobius(n/d)*k^d, d=divisors(n))/n)
    end:
T:= (n, k)-> add(b(n, k-j)*binomial(k,j)*(-1)^j, j=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[MoebiusMu[n/d]*k^d, {d, Divisors[n]}]/n]; T[n_, k_] := Sum[b[n, k-j]*Binomial[k, j]*(-1)^j, {j, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)
LyndonQ[q_]:=q==={}||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
allnorm[n_,k_]:=If[k===0,If[n===0,{{}}, {}],Join@@Permutations/@Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Select[Subsets[Range[n-1]+1],Length[#]===k-1&]];
Table[Length[Select[allnorm[n,k],LyndonQ]],{n,0,7},{k,0,n}] (* Gus Wiseman, Dec 22 2017 *)

T(n,k) = Sum_{j=0..k} (-1)^j * C(k,j) * A074650(n,k-j).

T(n,k) = Sum_{d|n} mu(d) * A087854(n/d, k) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Aug 20 2019

A273891 Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 18, 24, 12, 1, 11, 56, 136, 150, 60, 1, 16, 147, 612, 1200, 1080, 360, 1, 28, 411, 2619, 7905, 11970, 8820, 2520, 1, 44, 1084, 10480, 46400, 105840, 129360, 80640, 20160, 1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440

Offset: 1

Marko Riedel, Jun 02 2016

For bracelets, chiral pairs are counted as one.

			Triangle begins with T(1,1):
1;
1,  1;
1,  2,    1;
1,  4,    6,     3;
1,  6,   18,    24,     12;
1, 11,   56,   136,    150,     60;
1, 16,  147,   612,   1200,   1080,     360;
1, 28,  411,  2619,   7905,  11970,    8820,    2520;
1, 44, 1084, 10480,  46400, 105840,  129360,   80640,  20160;
1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440;
For T(4,2)=4, the arrangements are AAAB, AABB, ABAB, and ABBB, all achiral.
For T(4,4)=3, the arrangements are ABCD, ABDC, and ACBD, whose chiral partners are ADCB, ACDB, and ADBC respectively. - _Robert A. Russell_, Sep 26 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Marko Riedel, Maple code for A087854 and A273891.

Columns 1-6: A057427, A056342, A056343, A056344, A056345, A056346.
Row sums give A019537.
Cf. A087854 (oriented), A305540 (achiral), A305541 (chiral).

(* t = A081720 *) t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n+1)/2))/(2*n)]); T[n_, k_] := Sum[(-1)^i * Binomial[k, i]*t[n, k-i], {i, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)
Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,10}, {k,1,n}] // Flatten (* Robert A. Russell, Sep 26 2018 *)

T(n,k) = Sum_{i=0..k-1} (-1)^i * binomial(k,i) * A081720(n,k-i). - Andrew Howroyd, Mar 25 2017
From Robert A. Russell, Sep 26 2018: (Start)
T(n,k) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where S2 is the Stirling subset number A008277.
G.f. for column k>1: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j*x^d).
T(n,k) = (A087854(n,k) + A305540(n,k)) / 2 = A087854(n,k) - A305541(n,k) = A305541(n,k) + A305540(n,k).
(End)

A052823 A simple grammar: cycles of pairs of sequences.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 12, 18, 34, 58, 106, 186, 350, 630, 1180, 2190, 4114, 7710, 14600, 27594, 52486, 99878, 190744, 364722, 699250, 1342182, 2581426, 4971066, 9587578, 18512790, 35792566, 69273666, 134219794, 260301174, 505294126, 981706830, 1908881898

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Number of n-bead necklaces using exactly two different colors. - Robert A. Russell, Sep 26 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 788
Terry R. Payne, Luke Riley, Katie Atkinson, and Paul Dunne, Using Two Colour Necklaces to Fairly Allocate Coalition Value Calculations, 23rd IEEE/WIC Int'l Conf. Web Intel. Intel. Agent Tech. (WI-IAT 2024). See p. 7.
Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 2.

A000031 - 2.

Column k=2 of A087854.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Cycle(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

k=2; Prepend[Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}],0] (* Robert A. Russell, Sep 26 2018 *)

G.f.: Sum_{j>=1} phi(j)/j*log(-(x^j-1)^2/(2*x^j-1)).
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=2 is the number of colors and S2 is the Stirling subset number A008277. - Robert A. Russell, Sep 26 2018
a(n) ~ 2^n / n. - Vaclav Kotesovec, Sep 25 2019

More terms from Alois P. Heinz, Jan 25 2015

A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 3, 1, 6, 6, 1, 10, 21, 12, 1, 14, 36, 24, 1, 22, 93, 132, 60, 1, 30, 150, 240, 120, 1, 46, 345, 900, 960, 360, 1, 62, 540, 1560, 1800, 720, 1, 94, 1173, 4980, 9300, 7920, 2520, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 190, 3801, 24612, 71400, 103320, 73080, 20160, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320

Offset: 1

Robert A. Russell, Jun 04 2018

The number of achiral necklaces is equivalent to the number of achiral bracelets.

			The triangle begins with T(1,1):
1;
1,   1;
1,   2;
1,   4,     3;
1,   6,     6;
1,  10,    21,     12;
1,  14,    36,     24;
1,  22,    93,    132,     60;
1,  30,   150,    240,    120;
1,  46,   345,    900,    960,     360;
1,  62,   540,   1560,   1800,     720;
1,  94,  1173,   4980,   9300,    7920,    2520;
1, 126,  1806,   8400,  16800,   15120,    5040;
1, 190,  3801,  24612,  71400,  103320,   73080,   20160;
1, 254,  5796,  40824, 126000,  191520,  141120,   40320;
1, 382, 11973, 113652, 480060, 1048320, 1234800,  745920, 181440;
1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;
For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB.

Odd rows are A019538.
Even rows are A172106.
Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492.

Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] + StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 15}, {k, 1, Ceiling[(n + 1)/2]}] // Flatten

T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2));
tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 02 2018

T(n,k) = (k!/2) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), where S2(n,k) is the Stirling subset number A008277.
T(n,k) = 2*A273891(n,k) - A087854(n,k).
G.f. for column k>1: (k!/2) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2). - Robert A. Russell, Sep 26 2018

A305541 Triangle read by rows: T(n,k) is the number of chiral pairs of color loops of length n with exactly k different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 12, 24, 12, 0, 1, 35, 124, 150, 60, 0, 2, 111, 588, 1200, 1080, 360, 0, 6, 318, 2487, 7845, 11970, 8820, 2520, 0, 14, 934, 10240, 46280, 105840, 129360, 80640, 20160, 0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440, 0, 62, 7503, 158220, 1344900, 5873760, 14658840, 21772800, 19051200, 9072000, 1814400

Offset: 1

Robert A. Russell, Jun 04 2018

In other words, the number of n-bead bracelets with beads of exactly k different colors that when turned over are different from themselves. - Andrew Howroyd, Sep 13 2019

			Triangle T(n,k) begins:
  0;
  0,  0;
  0,  0,    1;
  0,  0,    3,     3;
  0,  0,   12,    24,     12;
  0,  1,   35,   124,    150,     60;
  0,  2,  111,   588,   1200,   1080,     360;
  0,  6,  318,  2487,   7845,  11970,    8820,    2520;
  0, 14,  934, 10240,  46280, 105840,  129360,   80640,  20160;
  0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440;
  ...
For T(4,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.
For T(4,4)=3, the chiral pairs are ABCD-ADCB, ABDC-ACDB, and ACBD-ADBC.

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

Columns 2-6 are A059076, A305542, A305543, A305544, and A305545.
Row sums are A326895.
Cf. A087854, A273891, A293496, A305540, A309651.

Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten

T(n,k) = {-k!*(stirling((n+1)\2,k,2) + stirling(n\2+1,k,2))/4 + k!*sumdiv(n,d, eulerphi(d)*stirling(n/d,k,2))/(2*n)} \\ Andrew Howroyd, Sep 13 2019

T(n,k) = -(k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2 n))*Sum_{d|n} phi(d)*S2(n/d,k), where S2(n,k) is the Stirling subset number A008277.
T(n,k) = A087854(n,k) - A273891(n,k).
T(n,k) = (A087854(n,k) - A305540(n,k)) / 2.
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293496(n, i). - Andrew Howroyd, Sep 13 2019

A054631 Triangle read by rows: row n (n >= 1) contains the numbers T(n,k) = Sum_{d|n} phi(d)*k^(n/d)/n, for k=1..n.

Original entry on oeis.org

1, 1, 3, 1, 4, 11, 1, 6, 24, 70, 1, 8, 51, 208, 629, 1, 14, 130, 700, 2635, 7826, 1, 20, 315, 2344, 11165, 39996, 117655, 1, 36, 834, 8230, 48915, 210126, 720916, 2097684, 1, 60, 2195, 29144, 217045, 1119796, 4483815, 14913200, 43046889

Offset: 1

N. J. A. Sloane, Apr 16 2000, revised Mar 21 2007

T(n,k) is the number of n-bead necklaces with up to k different colored beads. - Yves-Loic Martin, Sep 29 2020

			1;
1,  3;                                   (A000217)
1,  4,  11;                              (A006527)
1,  6,  24,   70;                        (A006528)
1,  8,  51,  208,   629;                 (A054620)
1, 14, 130,  700,  2635,  7826;          (A006565)
1, 20, 315, 2344, 11165, 39996, 117655;  (A054621)

Seiichi Manyama, Rows n = 1..140, flattened
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Cf. A054630, A054618, A054619, A087854. Lower triangle of A075195.

A054631 := proc(n,k) add( numtheory[phi](d)*k^(n/d),d=numtheory[divisors](n) ) ;  %/n ; end proc: # R. J. Mathar, Aug 30 2011

Needs["Combinatorica`"]; Table[Table[NumberOfNecklaces[n, k, Cyclic], {k, 1, n}], {n, 1, 8}] //Grid (* Geoffrey Critzer, Oct 07 2012, after code by T. D. Noe in A027671 *)
t[n_, k_] := Sum[EulerPhi[d]*k^(n/d)/n, {d, Divisors[n]}]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)

T(n, k) = sumdiv(n, d, eulerphi(d)*k^(n/d))/n; \\ Seiichi Manyama, Mar 10 2021

T(n, k) = sum(j=1, n, k^gcd(j, n))/n; \\ Seiichi Manyama, Mar 10 2021

T(n,k) = Sum_{j=1..k} binomial(k,j) * A087854(n, j). - Yves-Loic Martin, Sep 29 2020

T(n,k) = (1/n) * Sum_{j=1..n} k^gcd(j, n). - Seiichi Manyama, Mar 10 2021

A056283 Number of n-bead necklaces with exactly three different colored beads.

Original entry on oeis.org

0, 0, 2, 9, 30, 91, 258, 729, 2018, 5613, 15546, 43315, 120750, 338259, 950062, 2678499, 7573350, 21480739, 61088874, 174184755, 497812638, 1425847623, 4092087522, 11765822365, 33887517870, 97756387365, 282414624746, 816999710223, 2366509198350, 6862930841141

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=3, the two necklaces are ABC and ACB.

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]

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000031, A001867, A052823.

Column k=3 of A087854.

k=3; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = A001867(n) - 3*A000031(n) + 3.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=3 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=3 is the number of colors. (End)

A056284 Number of n-bead necklaces with exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 6, 48, 260, 1200, 5106, 20720, 81876, 318000, 1223136, 4675440, 17815020, 67769552, 257700906, 980240880, 3731753180, 14222737200, 54278580036, 207438938000, 793940475900, 3043140078000, 11681057249536, 44900438149296, 172824331826580, 666070256489680

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=4, the six necklaces are ABCD, ABDC, ACBD, ACDB, ADBC and ADCB.

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]

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001868.

Column k=4 of A087854.

k=4; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = my(k=4);(k!/n)*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Sep 27 2018

a(n) = A001868(n) - 4*A001867(n) + 6*A000031(n) - 4.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=4 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=4 is the number of colors. (End)

cp's OEIS Frontend

A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

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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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A254040 Number T(n,k) of primitive (= aperiodic) n-bead necklaces with colored beads of exactly k different colors; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A273891 Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads.

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A052823 A simple grammar: cycles of pairs of sequences.

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A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

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A305541 Triangle read by rows: T(n,k) is the number of chiral pairs of color loops of length n with exactly k different colors.

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