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 A084423 Set partitions up to rotations. 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS Franklin T. Adams-Watters and 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 FORMULA 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) EXAMPLE 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}}} MATHEMATICA <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 = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 14 2012, after Franklin T. Adams-Watters *) PROG (PARI) 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 (PARI) 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 CROSSREFS Cf. A080107, A084708, A152175. Sequence in context: A056293 A275311 A056294 * A068134 A249051 A329413 Adjacent sequences:  A084420 A084421 A084422 * A084424 A084425 A084426 KEYWORD nonn,nice AUTHOR Wouter Meeussen, Jun 26 2003 EXTENSIONS More terms from Robert G. Wilson v, Jun 27 2003 More terms from Franklin T. Adams-Watters, Jun 09 2008 STATUS approved

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Last modified October 2 18:23 EDT 2022. Contains 357228 sequences. (Running on oeis4.)