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A084708
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Number of set partitions up to rotations and reflections.
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8
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1, 2, 3, 7, 12, 37, 93, 354, 1350, 6351, 31950, 179307, 1071265, 6845581, 46162583, 327731950, 2437753740, 18948599220, 153498350745, 1293123243928, 11306475314467, 102425554299516, 959826755336242, 9290811905391501
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OFFSET
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1,2
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COMMENTS
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Equivalently, number of n-bead bracelets using any number of unlabeled (interchangable) colors. - Andrew Howroyd, Sep 25 2017
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LINKS
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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.
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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<<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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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