A084708 Number of set partitions up to rotations and reflections.

1, 2, 3, 7, 12, 37, 93, 354, 1350, 6351, 31950, 179307, 1071265, 6845581, 46162583, 327731950, 2437753740, 18948599220, 153498350745, 1293123243928, 11306475314467, 102425554299516, 959826755336242, 9290811905391501

Offset

1,2

Comments

Combines the symmetry operations of A080107 and A084423.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..577

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.

Tilman Piesk, Partition related number triangles

N. J. A. Sloane, Generating functions [From Wouter Meeussen, Dec 06 2008]

Formula

a(n) = (A080107(n)+A084423(n))/2. - Wouter Meeussen and Vladeta Jovovic, Nov 28 2008

Example

SetPartitions[6] is the first to decompose differently from A084423: 4 cycles of length 1, 2 of 2, 9 of 3, 16 of 6, 6 of 12.
a(7) = 1 + A056357(7) + A056358(7) + A056359(7) + A056360(7) + A056361(7) + 1 = 1 + 8 + 31 + 33 + 16 + 3 + 1 = 93.

Mathematica

<<DiscreteMath`NewCombinatorica`; (* see A080107 *); Table[{Length[ # ], First[ # ]}&/@ Split[Sort[Length/@Split[Sort[First[Sort[Flatten[ {#, Map[Sort, (#/. i_Integer:>w+1-i), 2]}& @(NestList[Sort[Sort/@(#/. i_Integer :> Mod[i+1, w, 1])]&, #, w]), 1]]]&/@SetPartitions[w]]]]], {w, 1, 10}]
u[0, j_]:=1; u[k_, j_]:=u[k, j]=Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; a[n_]:=1/n*Plus@@(EulerPhi[ # ]u[Quotient[n, # ], # ]&/@Divisors[n]); Table[a[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, {n, 40}] (* Wouter Meeussen, Dec 06 2008 *)

Crossrefs

Cf. A080107, A084423, A080510, A002872, A002874, A141003, A036075, A141004, A036077, A152176.

Sequence in context: A112837 A056355 A056356 * A035003 A143879 A056293

Adjacent sequences: A084705 A084706 A084707 * A084709 A084710 A084711

Keyword

nonn

Author

Wouter Meeussen, Jul 02 2003

Extensions

a(12) from Vladeta Jovovic, Jul 15 2007

More terms from Vladeta Jovovic's formula given in Mathematica line. - Wouter Meeussen, Dec 06 2008

Status

approved