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A008965 Number of necklaces of sets of beads containing a total of n beads. 43
1, 2, 3, 5, 7, 13, 19, 35, 59, 107, 187, 351, 631, 1181, 2191, 4115, 7711, 14601, 27595, 52487, 99879, 190745, 364723, 699251, 1342183, 2581427, 4971067, 9587579, 18512791, 35792567, 69273667, 134219795, 260301175, 505294127, 981706831, 1908881899, 3714566311, 7233642929 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

Equivalently, a(n) is the number of cyclic compositions of n. These could also be loosely described as cyclic partitions.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

R. Bekes, J. Pedersen and B. Shao, Mad tea party cyclic partitions, College Math. J., 43 (2012), 24-36.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

P. Flajolet and M. Soria, The Cycle Construction In SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 48

Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016) #16.8.2.

Index entries for sequences related to necklaces

FORMULA

a(n) = A000031(n-1) - 1.

G.f. sum(k>=1, phi(k)/k * log( 1/(1-B(x^k)) ) ) where B(x)=x/(1-x); see the Flajolet/Soria reference. - Joerg Arndt, Aug 06 2012

EXAMPLE

E.g. the 5 necklaces for n=4 are (3, 1), (4), (1, 1, 1, 1), (2, 1, 1), (2, 2).

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)).

For n=6 the 13 necklaces are

.1:  (1, 1, 1, 1, 1, 1),

.2:  (2, 1, 1, 1, 1),

.3:  (2, 1, 2, 1).

.4:  (2, 2, 1, 1),

.5:  (2, 2, 2),

.6:  (3, 1, 1, 1),

.7:  (3, 1, 2),

.8:  (3, 2, 1),

.9:  (3, 3),

10:  (4, 1, 1),

11:  (4, 2),

12:  (5, 1),

13:  (6).

[corrected by Marcel Vonk (mail(AT)marcelvonk.nl), Feb 05 2008]

MAPLE

with(combstruct): seq(combstruct[count]([ N, {N=Cycle(Set(Z, card>=1))}, unlabeled ], size=n), n=1..100);

MATHEMATICA

a[n_] := Sum[ EulerPhi[d]*2^(n/d), {d, Divisors[n]}]/n-1; Table[a[n], {n, 1, 38}] (* Jean-Fran├žois Alcover, Sep 04 2012, from A000031 *)

nn=35; Drop[Apply[Plus, Table[CoefficientList[Series[CycleIndex[ CyclicGroup[n], s]/.Table[s[i]->x^i/(1-x^i), {i, 1, n}], {x, 0, nn}], x], {n, 1, nn}]], 1]  (* Geoffrey Critzer, Oct 30 2012 *)

PROG

(PARI)

N=66;  x='x+O('x^N);

B(x)=x/(1-x);

A=sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k))));

Vec(A)

/* Joerg Arndt, Aug 06 2012 */

CROSSREFS

Row sums of A037306.

Cf. A000031.

Sequence in context: A138184 A236340 A273161 * A113864 A188754 A108310

Adjacent sequences:  A008962 A008963 A008964 * A008966 A008967 A008968

KEYWORD

nonn,easy,nice

AUTHOR

Paul Zimmermann

STATUS

approved

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Last modified October 20 06:43 EDT 2018. Contains 316378 sequences. (Running on oeis4.)