A203398 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.

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, 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, 2, 1, 2, 3, 0, 9, 0, 0, 0, 0, 0, 335

Offset: 1

Author

Geoffrey Critzer, Jan 01 2012

Keywords

• nonn
• tabl

Comments

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.

Examples

			  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  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
  2  1  2  3  0  9  0  0  0  0  0   335
		

Links

• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• Frank Ruskey, Combinatorial Generation Algorithm Algorithm 4.24, p. 95.

Crossrefs

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

Cf. A203399.

Programs

Mathematica

Needs["Combinatorica`"];
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}]]