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A084708 Number of set partitions up to rotations and reflections. 7
1, 2, 3, 7, 12, 37, 93, 354, 1350, 6351, 31950, 179307, 1071265, 6845581, 46162583, 327731950, 2437753740, 18948599220, 153498350745, 1293123243928, 11306475314467, 102425554299516, 959826755336242, 9290811905391501 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..24.

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

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)