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A261600 Number of primitive (aperiodic, or Lyndon) necklaces with n beads such that beads of a largest subset have label 0, beads of a largest remaining subset have label 1, and so on. 6
1, 1, 1, 3, 11, 49, 265, 1640, 11932, 96780, 887931, 8939050, 99298073, 1195617442, 15619180497, 219049941148, 3293800823995, 52746930894773, 897802366153076, 16167544246362566, 307372573010691195, 6148811682561388635, 129164845357775064609 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Eric Weisstein's World of Mathematics, Necklace

Wikipedia, Lyndon word

Wikipedia, Necklace (combinatorics)

Index entries for sequences related to necklaces

FORMULA

a(n) ~ c * (n-1)!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264818011615... . - Vaclav Kotesovec, Aug 27 2015

EXAMPLE

a(3) = 3: 001, 012, 021.

a(4) = 11: 0001, 0011, 0012, 0021, 0102, 0123, 0132, 0213, 0231, 0312, 0321.

MAPLE

with(numtheory):

b:= proc(n, i, g, d, j) option remember; `if`(g>0 and g<d, 0,

      `if`(n=0, `if`(d=g, 1, 0), `if`(i<1, 0, b(n, i-1, g, d, j)+

      `if`(i>n, 0, binomial(n/j, i/j)*b(n-i, i, igcd(i, g), d, j)))))

    end:

a:= n-> `if`(n=0, 1, add(add((f-> `if`(f=0, 0, f*b(n$2, 0, d, j)))(

                     mobius(j)), j=divisors(d)), d=divisors(n))/n):

seq(a(n), n=0..25);

MATHEMATICA

b[n_, i_, g_, d_, j_] := b[n, i, g, d, j] = If[g>0 && g<d, 0, If[n==0, If[d == g, 1, 0], If[i<1, 0, b[n, i-1, g, d, j] + If[i>n, 0, Binomial[n/j, i/j]*b[n-i, i, GCD[i, g], d, j]]]]]; a[n_] := If[n==0, 1, Sum[Sum[ Function[f, If[f==0, 0, f*b[n, n, 0, d, j]]][MoebiusMu[j]], {j, Divisors[ d]}], {d, Divisors[n]}]/n]; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Feb 22 2017, translated from Maple *)

CROSSREFS

Cf. A072605, A247551, A261531, A261599.

Sequence in context: A307030 A335788 A012316 * A331617 A193319 A265905

Adjacent sequences:  A261597 A261598 A261599 * A261601 A261602 A261603

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 27 2015

STATUS

approved

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Last modified June 24 00:08 EDT 2021. Contains 345403 sequences. (Running on oeis4.)