A032193 Number of necklaces with 8 black beads and n-8 white beads.

1, 1, 5, 15, 43, 99, 217, 429, 810, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30667, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328756, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922741

Offset: 8

Author

Christian G. Bower

Keywords

• nonn

Comments

The g.f. is Z(C_8,x)/x^8, the 8-variate cycle index polynomial for the cyclic group C_8, with substitution x[i]->1/(1-x^i), i=1,...,8. Therefore by Polya enumeration a(n+8) is the number of cyclically inequivalent 8-necklaces whose 8 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_8,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

From Petros Hadjicostas, Aug 31 2018: (Start)

The CIK[k] transform of sequence (c(n): n>=1) has generating function A_k(x) = (1/k)*Sum_{d|k} phi(d)*C(x^d)^{k/d}, where C(x) = Sum_{n>=1} c(n)*x^n is the g.f. of (c(n): n>=1).

When c(n) = 1 for all n >= 1, we get C(x) = x/(1-x) and A_k(x) = (x^k/k)*Sum_{d|k} phi(d)*(1-x^d)^{-k/d}, which is the g.f. of the number a_k(n) of necklaces of n beads of 2 colors with k of them black and n-k of them white.

Using Taylor expansions, we can easily prove that a_k(n) = (1/k)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d), which is Robert A. Russell's formula in the Mathematica code below.

For this sequence k = 8, and thus we get the formulae below.

(End)

Links

• Christian G. Bower, Transforms (2)
• David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
• Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Index entries for sequences related to necklaces
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Crossrefs

Column k=8 of A047996.

Cf. A004526, A005514, A007997, A008610, A008646, A032191, A032192.

Programs

Mathematica

k = 8; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[Series[1/8*(1/(1 - x)^8 + 1/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x^8)^1),{x, 0, 30}], x] (* Stefano Spezia, Sep 01 2018 *)

Formula

"CIK[ 8 ]" (necklace, indistinct, unlabeled, 8 parts) transform of 1, 1, 1, 1...

G.f.: (x^8)*(1-3*x+5*x^2+3*x^3-4*x^4+4*x^5+6*x^6-4*x^7+7*x^8-x^9+x^10+x^11)/((1-x)^4*(1-x^2)^2*(1-x^4)*(1-x^8)).

G.f.: 1/8*x^8*(1/(1-x)^8+1/(1-x^2)^4+2/(1-x^4)^2+4/(1-x^8)^1). - Herbert Kociemba, Oct 22 2016

a(n) = (1/8)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d - 1, 8/d - 1) = (1/n)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d, 8/d). - Petros Hadjicostas, Aug 31 2018