A084708 -id:A084708 - 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

A080107 Number of fixed points of permutation of SetPartitions under {1,2,...,n}->{n,n-1,...,1}. Number of symmetric arrangements of non-attacking rooks on upper half of n X n chessboard.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 31, 59, 164, 339, 999, 2210, 6841, 16033, 51790, 127643, 428131, 1103372, 3827967, 10269643, 36738144, 102225363, 376118747, 1082190554, 4086419601, 12126858113, 46910207114, 143268057587, 566845074703, 1778283994284, 7186474088735

Offset: 0

Wouter Meeussen, Mar 15 2003

nonn

Even-numbered terms a(2k) are A002872: 2,7,31,164,999 ("Sorting numbers"); odd-numbered terms are its binomial transform, A080337. The symmetrical set partitions of {-n,...,-1,0,1,...,n} can be classified by the partition containing 0. Thus we get the sum over k of {n choose k} times the number of symmetrical set partitions of 2n-2k elements. - Don Knuth, Nov 23 2003
Number of partitions of n numbers that are symmetrical and cannot be nested (i.e., include a pattern of the form abab). - Douglas Boffey, May 21 2015
Number of achiral color patterns in a row or loop of length n. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 23 2018
Also the number of self-complementary set partitions of {1, ..., n}. The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. - Gus Wiseman, Feb 13 2019

			Of the set partitions of 4, the following 7 are invariant under 1->4, 2->3, 3->2, 4->1: {{1,2,3,4}}, {{1,2},{3,4}}, {{1,4},{2,3}}, {{1,3},{2,4}}, {{1},{2,3},{4}}, {{1,4},{2},{3}}, {{1},{2},{3},{4}}, so a(4)=7.
For a(4)=7, the row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD (same as previous example).  The loop patterns are AAAA, AAAB, AABB, AABC, ABAB, ABAC, and ABCD. - _Robert A. Russell_, Apr 23 2018
From _Gus Wiseman_, Feb 13 2019: (Start)
The a(1) = 1 through a(5) = 12 self-complementary set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}      {{1245}{3}}
                   {{1}{2}{3}}  {{13}{24}}      {{135}{24}}
                                {{14}{23}}      {{15}{234}}
                                {{1}{23}{4}}    {{1}{234}{5}}
                                {{14}{2}{3}}    {{12}{3}{45}}
                                {{1}{2}{3}{4}}  {{135}{2}{4}}
                                                {{14}{25}{3}}
                                                {{15}{24}{3}}
                                                {{1}{24}{3}{5}}
                                                {{15}{2}{3}{4}}
                                                {{1}{2}{3}{4}{5}}
(End)

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 18.
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
Juan B. Gil and Luiz E. Lopez, Enumeration of symmetric arc diagrams, arXiv:2203.10589 [math.CO], 2022.
S. V. Pemmaraju and S. S. Skiena, The New Combinatorica, 2001.
Frank Ruskey, Set Partitions

Cf. A002872, A080337.
Cf. A125810. - Paul D. Hanna, Jan 19 2009
Cf. A000126, A000296, A124323, A169985, A324012, A324013, A324014.

< Range[n, 1, -1]]; t= 1 + RankSetPartition /@ t; t= ToCycles[t]; t= Cases[t, {_Integer}]; Length[t], {n, 7}]
(* second program: *)
QB[n_, q_] := QB[n, q] = Sum[QB[j, q] QBinomial[n-1, j, q], {j, 0, n-1}] // FunctionExpand // Simplify; QB[0, q_]=1; QB[1, q_]=1; Table[cc = CoefficientList[QB[n, q], q]; cc.Table[(-1)^(k+1), {k, 1, Length[cc]}], {n, 0, 30}] (* Jean-François Alcover, Feb 29 2016, after Paul D. Hanna *)
(* Ach[n, k] is the number of achiral color patterns for a row or loop of n
  colors containing exactly k different colors *)
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]]
Table[Sum[Ach[n, k], {k, 0, n}], {n, 0, 30}] (* Robert A. Russell, Apr 23 2018 *)
x[n_] := x[n] = If[n < 2, n+1, 2x[n-1] + (n-1)x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[Ceiling[n/2], k] x[k-Mod[n, 2]], {k, 0, Ceiling[n/2]}],
  {n, 0, 30}] (* Robert A. Russell, Apr 27 2018, after Knuth reference *)

Knuth gives recurrences and generating functions.
a(n) = Sum_{k=0..t(n)} (-1)^k*A125810(n,k) where A125810 is a triangle of coefficients for a q-analog of the Bell numbers and t(n)=A125811(n)-1. - Paul D. Hanna, Jan 19 2009
From Robert A. Russell, Apr 23 2018: (Start)
a(n) = Sum_{k=0..n} Ach(n,k) where
  Ach(n,k) = [n>1]*(k*Ach(n-2,k)+Ach(n-2,k-1)+Ach(n-2,k-2)) + [n<2]*[n==k]*[n>=0].
a(n) = 2*A103293(n+1) - A000110(n). (End)
a(n) = [n==0 mod 2]*Sum_{k=0..n/2} Stirling2(n/2, k)*A005425(k) + [n==1 mod 2] * Sum_{k=1..(n+1)/2} Stirling2((n+1)/2, k) * A005425(k-1). (from Knuth reference)
a(n) = 2*A084708(n) - A084423(n). - Robert A. Russell, Apr 27 2018

Offset set to 0 by Alois P. Heinz, May 23 2015

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

A320748 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented 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, 22, 9, 1, 1, 2, 3, 7, 12, 33, 40, 18, 1, 1, 2, 3, 7, 12, 36, 73, 100, 23, 1, 1, 2, 3, 7, 12, 37, 89, 237, 225, 44, 1, 1, 2, 3, 7, 12, 37, 92, 322, 703, 582, 63, 1, 1, 2, 3, 7, 12, 37, 93, 349, 1137, 2433, 1464, 122, 1, 1, 2, 3, 7, 12, 37, 93, 353, 1308, 4704, 8309, 3960, 190, 1, 1, 2, 3, 7, 12, 37, 93, 354, 1345, 5953, 19839, 30108, 10585, 362, 1

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted. An unoriented cycle counts each chiral pair as one, i.e., they are equivalent.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
T(n,k)=Pi_k(C_n) which is the number of non-equivalent partitions of the cycle on n vertices, with at most 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. - Bahman Ahmadi, Aug 21 2019
In other words, the number of n-bead bracelet 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   22    33    36     37     37     37     37     37     37     37 ...
1   9   40    73    89     92     93     93     93     93     93     93 ...
1  18  100   237   322    349    353    354    354    354    354    354 ...
1  23  225   703  1137   1308   1345   1349   1350   1350   1350   1350 ...
1  44  582  2433  4704   5953   6291   6345   6350   6351   6351   6351 ...
1  63 1464  8309 19839  28228  31284  31874  31944  31949  31950  31950 ...
1 122 3960 30108 88508 144587 171283 178190 179204 179300 179306 179307 ...
For T(7,2)=9, the patterns are AAAAAAB, AAAAABB, AAAABAB, AAAABBB, AAABAAB, AAABABB, AABAABB, AABABAB, and AAABABB; only the last is chiral, paired with AAABBAB.

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.

Partial row sums of A152176.
Columns 1-6 are A057427, A000011, A056353, A056354, A056355, A056356
For increasing k, columns converge to A084708.
Cf. A320747 (oriented), 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]]
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) - A320742(n,k) = A320742(n,k) + A305749(n,k).

A320749 Number of chiral pairs of color patterns (set partitions) in a cycle of length n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 190, 1011, 5352, 29740, 172466, 1055232, 6793791, 46034940, 327303819, 2436650368, 18944771253, 153488081102, 1293086505784, 11306373089104, 102425178180769, 959825673145688, 9290807818971900, 92771800581171418, 954447025978145744, 10105871186441842623, 110009631951698573068, 1229996584263621368224, 14112483571723367245825, 166021918475962174194914, 2001010469483653602192695

Offset: 1

Robert A. Russell, Oct 22 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.

			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.

Row sums of A320647.
Columns of A320742 converge to this as k increases.
Cf. A084423 (oriented), A084708 (unoriented), A080107 (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]]
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j,n}], {n,40}]

a(n) = Sum_{j=1..n} -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)).

a(n) = (A084423(n) - A080107(n)) / 2 = A084423(n) - A084708(n) = A084708(n) - A080107(n).

A276548 Number of primitive (period n) bracelet structures using an infinite alphabet.

Original entry on oeis.org

1, 1, 2, 5, 11, 33, 92, 347, 1347, 6338, 31949, 179265, 1071264, 6845487, 46162569, 327731596, 2437753739, 18948597836, 153498350744, 1293123237572, 11306475314372, 102425554267565, 959826755336241, 9290811905211847

Offset: 1

Andrew Howroyd, Apr 09 2017

nonn

Row sums of A276543.

Cf. A084708.

u[0, ] = 1; u[k, j_] := u[k, j] = Sum[Binomial[k - 1, i - 1] Total[u[k - i, j] #^(i - 1) & /@ Divisors[j]], {i, k}];
b[n_] := 1/n*Total[EulerPhi[#] u[Quotient[n, #], #]& /@ Divisors[n] ];
A084708[n_] := b[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;
a[n_] := Sum[MoebiusMu[n/d]*A084708[d], {d, Divisors[n]}];
Array[a, 24] (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd and Wouter Meeussen *)

a(n) = Sum_{d|n} mu(n/d) * A084708(d).

A211354 Refined triangle A211358: T(n,k) is the number of partitions up to rotation and reflection of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 3, 8, 3, 6, 1, 5, 3, 3, 1, 1, 3, 4, 12, 4, 18, 3, 11, 9, 7, 1, 12, 3, 4, 1, 1, 4, 5, 22, 8, 38, 5, 39, 33, 25, 4, 57, 12, 19, 1, 17, 22, 25, 4, 5, 7, 1, 1, 4, 7, 30, 10, 76, 10, 85, 76, 55

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,7,12,37... (A084708)
Row maxima: 1,1,1,2,3,8,18,57,228,668,3220
Distinct entries per row: 1,1,1,2,3,5,8,14,17,26,30
Rightmost columns are those from the triangle A052307  without the second column.

Tilman Piesk, Rows n=1..11 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211358, A000041, A084708, A052307.

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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A080107 Number of fixed points of permutation of SetPartitions under {1,2,...,n}->{n,n-1,...,1}. Number of symmetric arrangements of non-attacking rooks on upper half of n X n chessboard.

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A084423 Set partitions up to rotations.

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A320748 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented cycle of length n using k or fewer colors (subsets).

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A320749 Number of chiral pairs of color patterns (set partitions) in a cycle of length n.

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A276548 Number of primitive (period n) bracelet structures using an infinite alphabet.

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A211354 Refined triangle A211358: T(n,k) is the number of partitions up to rotation and reflection of an n-set that are of type k (k-th integer partition, defined by A194602).

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