A056292 -id:A056292 - 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

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

A056297 Number of n-bead necklace structures using exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 1, 2, 13, 50, 221, 866, 3437, 13250, 51075, 194810, 742651, 2823766, 10738881, 40843370, 155494751, 592614050, 2261625725, 8643289534, 33080920607, 126797503250, 486710971595, 1870851589554, 7201014763285, 27752927359726, 107092397450897

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]

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 4 of A152175.

Cf. A002076, A056284, A056292.

From Robert A. Russell, May 29 2018: (Start)
Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &], Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 4], {n, 1, 40}] (* after Gilbert and Riordan *)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[ Divisible[#,12], StirlingS2[n/#+3,4] - 3 StirlingS2[n/#+2,4] + 2 StirlingS2[n/#+1,4], Divisible[#,6], 3 StirlingS2[n/#+2,4] - 9 StirlingS2[n/#+1,4] + 6 StirlingS2[n/#,4], Divisible[#,4], StirlingS2[n/#+3,4] - 5 StirlingS2[n/#+2,4] + 10 StirlingS2[n/#+1,4] - 8 StirlingS2[n/#,4], Divisible[#,3], 2 StirlingS2[n/#+2,4] - 8 StirlingS2[n/#+1,4] + 9 StirlingS2[n/#,4], Divisible[#,2], StirlingS2[n/#+2,4] - StirlingS2[n/#+1,4] - 2 StirlingS2[n/#,4], True, StirlingS2[n/#,4]] &],{n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 12], Log[1-4x^d] - Log[1-3x^d], Divisible[d, 6], (3 Log[1-4x^d] - 4 Log[1-3x^d]) / 4, Divisible[d, 4], (2 Log[1-4x^d] - 2 Log[1-3x^d] + Log[1-x^d]) / 3, Divisible[d, 3], (3 Log[1-4x^d] - 4 Log[1-3x^d] + 2 Log[1-2x^d] - 4 Log[1-x^d]) / 8, Divisible[d, 2], (5 Log[1-4x^d] - 8 Log[1-3x^d] + 4 Log[1-x^d]) / 12, True, (Log[1-4x^d] - 4 Log[1-3x^d] + 6 Log[1-2x^d] - 4 Log[1-x^d]) / 24], {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

a(n) = A056292(n) - A002076(n).
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 12] * (S2(n/d + 3, 4) - 3*S2(n/d+2,4) + 2*S2(n/d + 1, 4)) + [d==6 mod 12] * (3*S2(n/d + 2, 4) - 9*S2(n/d + 1, 4) + 3*S2(n/d, 4)) + [d==4 mod 12 | d==8 mod 12] * (S2(n/d + 3, 4) - 5*S2(n/d + 2, 4) - 10*S2(n/d + 1, 4) - 8*S2(n/d, 4)) + [d==3 mod 12 | d=9 mod 12] * (2*S2(n/d + 2, 4) - 8*S2(n/d + 1, 4) - 2*S2(n/d,4)) + [d==2 mod 12 | d==10 mod 12] * (S2(n/d + 2, 4) - S2(n/d + 1, 4) + 9*S2(n/d, 4)) + [d mod 12 in {1,5,7,11}] * S2(n/d, 4)), where S2(n,k) is the Stirling subset number, A008277.
G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 12] * (log(1-4x^d) - log(1-3x^d)) +[d==6 mod 12] * (3*log(1-4x^d) - 4*log(1-3x^d)) / 4 + [d==4 mod 12 | d==8 mod 12] * (2*log(1-4x^d) - 2*log(1-3x^d) + log(1-x^d)) / 3 + [d==3 mod 12 | d==9 mod 12] * (3*log(1-4x^d) - 4*log(1-3x^d) + 2*log(1-2x^d) - 4*log(1-x^d)) / 8 + [d==2 mod 12 | d=10 mod 12] * (5*log(1-4x^d) - 8*log(1-3x^d) + 4*log(1-x^d)) / 12 + [d mod 12 in {1,5,7,11}] * (log(1-4x^d) - 4*log(1-3x^d) + 6*log(1-2x^d) - 4*log(1-x^d)) / 24).
(End)

A305750 Number of achiral color patterns (set partitions) in a row or cycle of length n with 4 or fewer colors (subsets).

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 27, 43, 107, 171, 427, 683, 1707, 2731, 6827, 10923, 27307, 43691, 109227, 174763, 436907, 699051, 1747627, 2796203, 6990507, 11184811, 27962027, 44739243, 111848107, 178956971, 447392427, 715827883, 1789569707, 2863311531, 7158278827

Offset: 0

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCD are equivalent, as are AABCD and BBCDA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABBCD = BBCDAA = CDAABB.

			For a(4) = 7, the achiral row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD. The cycle patterns are AAAA, AAAB, AABB, ABAB, AABC, ABAC, and ABCD.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Fourth column of A305749.
Cf. A124303 (oriented), A056323 (unoriented), A320934 (chiral), for rows.
Cf. A056292 (oriented), A056354 (unoriented), A320744 (chiral), for cycles.

a:=[1,2,3];; for n in [4..40] do a[n]:=a[n-1]+4*a[n-2]-4*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 28 2018

seq(coeff(series((1-3*x^2+x^3)/((1-x)*(1-4*x^2)),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 28 2018

Table[If[EvenQ[n], StirlingS2[(n+8)/2, 4] - 8 StirlingS2[(n+6)/2, 4] + 22 StirlingS2[(n+4)/2, 4] - 23 StirlingS2[(n+2)/2, 4] + 6 StirlingS2[n/2, 4], StirlingS2[(n+7)/2, 4] - 7 StirlingS2[(n+5)/2, 4] + 16 StirlingS2[(n+3)/2, 4] - 12 StirlingS2[(n+1)/2, 4]], {n, 0, 40}]
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 *)
k=4; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]
(* or *)
CoefficientList[Series[(1-3x^2+x^3)/((1-x)(1-4x^2)), {x,0,40}], x]
(* or *)
Join[{1},LinearRecurrence[{1,4,-4},{1,2,3},40]]
(* or *)
Join[{1},Table[If[EvenQ[n], (4+5 4^(n/2))/12, (2+4^((n+1)/2))/6], {n,40}]]

a(n) = Sum_{j=0..4} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].
G.f.: (1 - 3x^2 + x^3) / ((1-x) * (1-4x^2)).
a(2m) = S2(m+4,4) - 8*S2(m+3,4) + 22*S2(m+2,4) - 23*S2(m+1,4) + 6*S2(m,4);
a(2m-1) = S2(m+3,4) - 7*S2(m+2,4) + 16*S2(m+1,4) - 12*S2(m,4), where S2(n,k) is the Stirling subset number A008277.
For m>0, a(2m) = (4 + 5*4^m) / 12.
a(2m-1) = (2 + 4^m) / 6.
a(n) = 2*A056323(n) - A124303(n) = A124303(n) - 2*A320934(n) = A056323(n) - A320934(n).
a(n) = 2*A056354(n) - A056292(n) = A056292(n) - 2*A320744(n) = A056354(n) - A320744(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3). - Muniru A Asiru, Oct 28 2018

A056298 Number of n-bead necklace structures using exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 20, 136, 773, 4281, 22430, 115100, 577577, 2863227, 14051164, 68515514, 332514803, 1608800691, 7767857090, 37460388596, 180536313547, 869901397479, 4192038616700, 20208367895980

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]

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 5 of A152175.

Cf. A056285, A056292, A056293.

From Robert A. Russell, May 29 2018: (Start)
Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &],
  Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 5],
  {n, 1, 40}] (* after Gilbert and Riordan *)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+4, 5] - 6 StirlingS2[n/#+3, 5] + 11 StirlingS2[n/#+2, 5] - 6 StirlingS2[n/#+1, 5], Divisible[#, 30], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 26 StirlingS2[n/#+2, 5] - 43 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 20], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 23 StirlingS2[n/#+2, 5] - 24 StirlingS2[n/#+1, 5], Divisible[#, 15], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 65 StirlingS2[n/#+1, 5] + 45 StirlingS2[n/#, 5], Divisible[#, 12], 4 StirlingS2[n/#+3, 5] - 24 StirlingS2[n/#+2, 5] + 44 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 10], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 61 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 6], 2 StirlingS2[n/#+3, 5] - 9 StirlingS2[n/#+2, 5] + 7 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], Divisible[#, 5], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 35 StirlingS2[n/#+2, 5] - 50 StirlingS2[n/#+1, 5] + 25 StirlingS2[n/#, 5], Divisible[#, 4], 2 StirlingS2[n/#+3, 5] - 12 StirlingS2[n/#+2, 5] + 26 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 3], 3 StirlingS2[n/#+2, 5] - 15 StirlingS2[n/#+1, 5] + 21 StirlingS2[n/#, 5], Divisible[#, 2], 3 StirlingS2[n/#+2, 5] - 11 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], True, StirlingS2[n/#, 5]] &], {n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[
  Divisible[d, 60], Log[1 - 5x^d] - Log[1 - 4x^d], Divisible[d, 30],
  (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + Log[1 - x^d]) / 4, Divisible[d, 20],
  (2 Log[1 - 5x^d] - 2 Log[1 - 4x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 3,
  Divisible[d, 15], (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + 2 Log[1 - 3x^d] -
  2 Log[1 - 2x^d] + 3 Log[1 - x^d]) / 8, Divisible[d, 12],
  (4 Log[1 - 5x^d] - 5 Log[1 - 4x^d]) / 5, Divisible[d, 10],
  (5 Log[1 - 5x^d] - 5 Log[1 - 4x^d] + 4 Log[1 - 2x^d] - Log[1 - x^d]) / 12,
  Divisible[d, 6], (11 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 5 Log[1 - x^d]) /
  20, Divisible[d, 5], (5 Log[1 - 5x^d] - Log[1 - 4x^d] + 2 Log[1 - 3x^d] -
  2 Log[1 - 2x^d] + Log[1 - x^d]) / 24, Divisible[d, 4], (7 Log[1 - 5x^d] -
  10 Log[1 - 4x^d] + 5 Log[1 - 2x^d] - 5 Log[1 - x^d]) / 15,
  Divisible[d, 3], (7 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 10 Log[1 - 3x^d] -
  10 Log[1 - 2x^d] + 15 Log[1 - x^d]) / 40, Divisible[d, 2],
  (13 Log[1 - 5x^d] - 25 Log[1 - 4x^d] + 20 Log[1 - 2x^d] -
  5 Log[1 - x^d]) / 60, True, (Log[1 - 5x^d] - 5 Log[1 - 4x^d] +
  10 Log[1 - 3x^d] - 10 Log[1 - 2x^d] + 5 Log[1 - x^d]) / 120], {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

a(n) = A056293(n) - A056292(n).
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (S2(n/d+4,5) -
  6*S2(n/d+3,5) + 11*S2(n/d+2,5) - 6*S2(n/d+1,5)) + [d==30 mod 60] *
  (S2(n/d+4,5) - 8*S2(n/d+3,5) + 26*S2(n/d+2,5) - 43*S2(n/d+1,5) +
  30*S2(n/d,5)) + [d==20 mod 60 | d==40 mod 60] * (S2(n/d+4,5) -
  8*S2(n/d+3,5) + 23*S2(n/d+2,5) - 24*S2(n/d+1,5)) + [d==15 mod 60 |
  d==45 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) + 38*S2(n/d+2,5) -
  65*S2(n/d+1,5) + 45*S2(n/d,5)) + [d mod 60 in {12,24,36,48}] *
  (4*S2(n/d+3,5) - 24*S2(n/d+2,5) + 44*S2(n/d+1,5) - 24*S2(n/d,5)) +
  [d=10 mod 60 | d==50 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) +
  38*S2(n/d+2,5) - 61*S2(n/d+1,5) + 30*S2(n/d,5)) + [d mod 60 in
  {6,18,42,54}] * (2*S2(n/d+3,5) - 9*S2(n/d+2,5) + 7*S2(n/d+1,5) +
  6*S2(n/d,5)) + [d mod 60 in {5,25,35,55}] * (S2(n/d+4,5) -
  10*S2(n/d+3,5) + 35*S2(n/d+2,5) - 50*S2(n/d+1,5) + 25*S2(n/d,5)) +
  [d mod 60 in {4,8,16,28,32,44,52,56}] * (2*S2(n/d+3,5) - 12*S2(n/d+2,5) +
  26*S2(n/d+1,5) - 24*S2(n/d,5)) + [d mod 60 in {3,9,21,27,33,39,51,57}] *
  (3*S2(n/d+2,5) - 15*S2(n/d+1,5) + 21*S2(n/d,5)) + [d mod 60 in
  {2,14,22,26,34,38,46,58}] * (3*S2(n/d+2,5) - 11*S2(n/d+1,5) +
  6*S2(n/d,5)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,
  59}] * S2(n/d,5)), where S2(n,k) is the Stirling subset number, A008277.
G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-4x^d) -
  log(1-3x^d)) + [d==30 mod 60] * (3*log[1-5x^d) - 3*log(1-4x^d) +
  log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (2*log(1-5x^d) -
  2*log(1-4x^d) + log(1-2x^d) - log(1-x^d)) / 3 +
  [d==15 mod 60 | d==45 mod 60] * (3*log(1-5x^d) - 3*log(1-4x^d) +
  2*log(1-3x^d) - 2*log(1-2x^d) + 3*log(1-x^d)) / 8 + [d mod 60 in
  {12,24,36,48}] * (4*log(1-5x^d) - 5*log(1-4x^d)) / 5 + [d=10 mod 60 |
  d==50 mod 60] * (5*log(1-5x^d) - 5*log(1-4x^d) + 4*log(1-2x^d) -
  log(1-x^d)) / 12 + [d mod 60 in {6,18,42,54}] * (11*log(1-5x^d) -
  15*log(1-4x^d) + 5*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *
  (5*log(1-5x^d) - log(1-4x^d) + 2*log(1-3x^d) - 2*log(1-2x^d) +
  log(1-x^d)) / 24 + [d mod 60 in {4,8,16,28,32,44,52,56}] *
  (7*log(1-5x^d) - 10*log(1-4x^d) + 5*log(1-2x^d) - 5*log(1-x^d)) /
  15 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (7*log(1-5x^d) -
  15*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 15*log(1-x^d)) /
  40 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (13*log(1-5x^d) -
  25*log(1-4x^d) + 20*log(1-2x^d) - 5*log(1-x^d)) / 60 + [d mod 60 in
  {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (log(1-5x^d) -
  5*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 5*log(1-x^d)) / 120).
(End)

A056300 Number of primitive (period n) n-bead necklace structures using a maximum of four different colored beads.

Original entry on oeis.org

1, 1, 2, 5, 10, 35, 102, 360, 1232, 4427, 15934, 58465, 215250, 799593, 2983204, 11187200, 42109450, 159081482, 602809326, 2290679807, 8726308212, 33318645341, 127479700198, 488672244040, 1876500180280

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]

Cf. A008683, A027377, A056292.

a(n) = Sum_{d|n} mu(d) * A056292(n/d); mu = A008683.

A309673 Number of n-bead necklace structures using a maximum of four different colored beads and no adjacent beads having the same color.

Original entry on oeis.org

0, 1, 1, 3, 2, 9, 13, 41, 94, 257, 671, 1881, 5110, 14301, 39871, 112281, 316520, 897297, 2548819, 7265383, 20754748, 59437181, 170549237, 490338539, 1412147684, 4073528481, 11767897903, 34042917197, 98606864030, 285960106473, 830206177801, 2412787265021

Offset: 1

Andrew Howroyd, Oct 05 2019

nonn

Colors may be permuted without changing the necklace structure.

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

Cf. A056292, A306888, A327396, A328130.

a(n) = Sum_{k=1..4} A327396(n, k).

Terms a(24) and beyond from Andrew Howroyd, Oct 10 2019

A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 30, 130, 532, 2006, 7626, 28401, 106260, 396435, 1486147, 5580130, 21032880, 79486763, 301317282, 1145123672, 4362804633, 16658456825, 63738451998, 244332656201, 938244497740, 3608640426930, 13899977105315, 53614228550220, 207061964668740, 800639722002163, 3099251007215286, 12009598156277090, 46582685655751645, 180850428684482360

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.
There are nonrecursive formulas, generating functions, and computer programs for A056292 and A305750, which can be used in conjunction with the first formula.

			For a(6)=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..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 4 of A320742.

Cf. A056292 (oriented), A056354 (unoriented), A305750 (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 *)
k=4; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A056292(n) - A305750(n)) / 2 = A056292(n) - A056354(n) = A056354(n) - A305750(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=4 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)).
a(n) = A059053(n) + A320643(n) + A320644(n).

A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 26, 10, 1, 1, 2, 3, 7, 12, 39, 53, 20, 1, 1, 2, 3, 7, 12, 42, 103, 146, 30, 1, 1, 2, 3, 7, 12, 43, 123, 367, 369, 56, 1, 1, 2, 3, 7, 12, 43, 126, 503, 1235, 1002, 94, 1, 1, 2, 3, 7, 12, 43, 127, 539, 2008, 4439, 2685, 180, 1, 1, 2, 3, 7, 12, 43, 127, 543, 2304, 8720, 15935, 7434, 316, 1, 1, 2, 3, 7, 12, 43, 127, 544, 2356, 11023, 38365, 58509, 20441, 596, 1

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted. An oriented cycle counts each chiral pair as two.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
In other words, the number of n-bead necklace structures using a maximum of k different colored beads. - Andrew Howroyd, Oct 30 2019

			Array begins with T(1,1):
1   1    1     1      1      1      1      1      1      1      1      1 ...
1   2    2     2      2      2      2      2      2      2      2      2 ...
1   2    3     3      3      3      3      3      3      3      3      3 ...
1   4    6     7      7      7      7      7      7      7      7      7 ...
1   4    9    11     12     12     12     12     12     12     12     12 ...
1   8   26    39     42     43     43     43     43     43     43     43 ...
1  10   53   103    123    126    127    127    127    127    127    127 ...
1  20  146   367    503    539    543    544    544    544    544    544 ...
1  30  369  1235   2008   2304   2356   2360   2361   2361   2361   2361 ...
1  56 1002  4439   8720  11023  11619  11697  11702  11703  11703  11703 ...
1  94 2685 15935  38365  54682  60499  61579  61684  61689  61690  61690 ...
1 180 7434 58509 173609 284071 336447 349746 351619 351766 351772 351773 ...
For T(4,2)=4, the patterns are AAAA, AAAB, AABB, and ABAB.
For T(4,3)=6, the patterns are the above four, 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.

Partial row sums of A152175.
Columns 1-6 are A057427, A000013, A002076, A056292, A056293, A056294.
For increasing k, columns converge to A084423.
Cf. A320748 (unoriented), A320742 (chiral), 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]]
Table[Sum[DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j] &], {j,k-n+1}]/n, {k,15}, {n,k}] // Flatten

\\ R 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)); 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) = (1/n)*Sum_{j=1..k} Sum_{d|n} phi(d)*A(d,n/d,j), 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)).

T(n,k) = A320748(n,k) + A320742(n,k) = 2*A320748(n,k) - A305749(n,k) = A305749(n,k) + 2*A320742(n,k).

A328743 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using a maximum of four different colored beads.

Original entry on oeis.org

1, 1, 0, 1, 2, 10, 26, 102, 336, 1225, 4352, 15934, 58190, 215250, 798720, 2983136, 11184128, 42109450, 159070352, 602809326, 2290640486, 8726307432, 33318502400, 127479700198, 488671717040, 1876500180280, 7217306664960, 27799998938929, 107228560958610, 414124108294450

Offset: 0

Andrew Howroyd, Oct 26 2019

nonn

Permuting the colors does not change the structure.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A056292, A056300, A327693, A328741.

a(n) = Sum_{k=1..4} A327693(n, k) for n > 0.

a(p) = A056300(p) for prime p >= 5.

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A056297 Number of n-bead necklace structures using exactly four different colored beads.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A305750 Number of achiral color patterns (set partitions) in a row or cycle of length n with 4 or fewer colors (subsets).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Formula

A056298 Number of n-bead necklace structures using exactly five different colored beads.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A056300 Number of primitive (period n) n-bead necklace structures using a maximum of four different colored beads.

Views

Author

Keywords

Comments

References

Crossrefs

Formula

A309673 Number of n-bead necklace structures using a maximum of four different colored beads and no adjacent beads having the same color.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

Views

Author