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Search: id:A019536
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| A019536 |
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a(n) = number of necklaces of n beads with up to n unlabeled colors. |
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+0
2
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| 1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, 147512053, 2340964372, 40527565261, 760095929840, 15352212731933, 332228417657960, 7668868648772701, 188085259070219000, 4884294069438337429
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Moebius transform of a(n) equals A060223. Possibly identical to A019536.
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LINKS
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M. Goebel, On the number of special permutation-invariant orbits and terms, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)
F. Ruskey, Necklaces with Fixed Density
Eric Weisstein's world of Mathematics, Necklaces
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FORMULA
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See Mma code
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EXAMPLE
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a(3)=5 since the partitions of the 3 (unlabeled) colors are {3}:RRR, {2,1}:RRG, and {1,1,1}:RGB, with multiplicities resp. *1, *2, *1, ( *2 since partitioning 3 beads over a bin of 2 and a bin of 1), so the necklaces are {R,R,R},{R,R,G},{R,G,G},{R,G,B},{R,B,G}.
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MATHEMATICA
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Needs["DiscreteMath`Combinatorica`"]; mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]]; neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li]; Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n]; Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
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CROSSREFS
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Cf. A000670.
Row sums of A087854. - DELEHAM Philippe.
Adjacent sequences: A019533 A019534 A019535 this_sequence A019537 A019538 A019539
Sequence in context: A009551 A006924 A006867 this_sequence A129949 A127065 A052850
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KEYWORD
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easy,nonn
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AUTHOR
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Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)
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EXTENSIONS
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Edited by Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 06 2002
Corrected by T. D. Noe (noe(AT)sspectra.com), Oct 31 2006
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