A261599 Number of primitive (aperiodic, or Lyndon) necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

1, 1, 0, 1, 1, 3, 13, 24, 67, 252, 1795, 4038, 16812, 61750, 349806, 3485026, 10391070, 49433135, 240064988, 1282012986, 9167581934, 131550811985, 459677212302, 2707382738558, 14318807586215, 94084166753923, 601900541189696, 5894253303715121

Offset: 0

Author

Alois P. Heinz, Aug 25 2015

Keywords

• nonn

Examples

			a(4) = 1: 0001.
a(5) = 3: 00001, 00011, 00101.
a(6) = 13: 000001, 000011, 000101, 000112, 000121, 000122, 001012, 001021, 001022, 001102, 001201, 001202, 010102.
a(7) = 24: 0000001, 0000011, 0000101, 0000111, 0000112, 0000121, 0000122, 0001001, 0001011, 0001012, 0001021, 0001022, 0001101, 0001102, 0001201, 0001202, 0010011, 0010012, 0010021, 0010022, 0010101, 0010102, 0010201, 0010202.
		

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

Crossrefs

Cf. A007837, A261531, A261600.

Programs

Maple

with(numtheory):
b:= proc(n, i, g, d, j) option remember; `if`(i*(i+1)/20
       and gn, 0, binomial(n/j, i/j)*b(n-i, i-1, 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..30);

Mathematica

a[0] = 1; a[n_] := With[{P = Product[1 + x^k/k!, {k, 1, n}] + O[x]^(n+1) // Normal}, DivisorSum[n, MoebiusMu[n/#]*#!*Coefficient[P, x, #]&]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, after Andrew Howroyd *)

PARI

a(n)={if(n==0, 1, my(p=prod(k=1, n, (1+x^k/k!) + O(x*x^n))); sumdiv(n, d, moebius(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017

Formula

a(n) = (1/n) * Sum_{d | n} moebius(n/d) * A007837(d) for n>0. - Andrew Howroyd, Dec 21 2017