A203398 - OEIS

%I #21 Jun 07 2019 19:39:16

%S 2,2,1,2,0,2,2,1,0,3,2,0,0,0,6,2,1,2,0,0,9,2,0,0,0,0,0,18,2,1,0,3,0,0,

%T 0,30,2,0,2,0,0,0,0,0,56,2,1,0,0,6,0,0,0,0,99,2,0,0,0,0,0,0,0,0,0,186,

%U 2,1,2,3,0,9,0,0,0,0,0,335

%N T(n,k), a triangular array read by rows, is the number of classes of equivalent 2-color n-bead necklaces (turning over is not allowed) that have k necklaces.

%C Equivalently, the cyclic group of order n acts on the set of length n binary sequences.  T(n,k) is the number of orbits that have k elements.

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H Frank Ruskey, <a href="https://web.archive.org/web/20171022155546/http://www.1stworks.com/ref/RuskeyCombGen.pdf">Combinatorial Generation Algorithm Algorithm 4.24, p. 95</a>.

%e   2

%e   2  1

%e   2  0  2

%e   2  1  0  3

%e   2  0  0  0  6

%e   2  1  2  0  0  9

%e   2  0  0  0  0  0  18

%e   2  1  0  3  0  0  0  30

%e   2  0  2  0  0  0  0  0  56

%e   2  1  0  0  6  0  0  0  0  99

%e   2  0  0  0  0  0  0  0  0  0  186

%e   2  1  2  3  0  9  0  0  0  0  0   335

%t Needs["Combinatorica`"];

%t f[list_] := Sort[NestList[RotateLeft, list, Length[list]-1]]; Flatten[Table[Distribution[Map[Length, Map[Union, Union[Map[f, Strings[{0, 1}, n]]]]], Range[n]], {n, 1, 12}]]

%Y A000031 (row sums), T(n,n) = A001037, T(n,n) = A064535 when n is prime, T(n,k) = A001037(k) when k divides n.

%Y Cf. A203399.

%K nonn,tabl

%O 1,1

%A _Geoffrey Critzer_, Jan 01 2012