Search: id:A008965
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%I A008965
%S A008965 1,2,3,5,7,13,19,35,59,107,187,351,631,1181,2191,4115,7711,
%T A008965 14601,27595,52487,99879,190745,364723,699251,1342183,2581427,
%U A008965 4971067,9587579,18512791,35792567,69273667,134219795,260301175
%N A008965 Number of necklaces of sets of beads containing a total of n beads.
%C A008965 A necklace of sets of beads is a cycle where each element of the cycle 
               is itself a set of beads, the total size being the total number of 
               beads.
%C A008965 Equivalently, a(n) is the number of cyclic compositions of n.
%H A008965 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A008965 <a href="Sindx_Ne.html#necklaces">Index entries for sequences related 
               to necklaces</a>
%H A008965 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               48
%e A008965 E.g. the 5 necklaces for n=4 are (3, 1), (4), (1, 1, 1, 1), (2, 1, 1), 
               (2, 2).
%e A008965 In the Combstruct language these can be described as Cycle(Set(Z), Set(Z), 
               Set(Z), Set(Z)), Cycle(Set(Z, Z), Set(Z), Set(Z)), Cycle(Set(Z, Z, 
               Z, Z)), Cycle(Set(Z, Z), Set(Z, Z)), Cycle(Set(Z), Set(Z, Z, Z)).
%e A008965 For n=6 the 13 necklaces are (2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 
               4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 
               1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1). [Example corrected by Marcel 
               Vonk (mail(AT)marcelvonk.nl), Feb 05 2008]
%p A008965 with(combstruct): seq(combstruct[count]([ N,{N=Cycle(Set(Z,card>=1))},
               unlabeled ],size=n), n=1..100);
%Y A008965 a(n) = A000031(n) - 1 allowing different offsets.
%Y A008965 Sequence in context: A075580 A077132 A138184 this_sequence A113864 A108310 
               A146999
%Y A008965 Adjacent sequences: A008962 A008963 A008964 this_sequence A008966 A008967 
               A008968
%K A008965 nonn,easy,nice
%O A008965 1,2
%A A008965 Paul.Zimmermann(AT)loria.fr

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