A323872 Number of n X n aperiodic binary toroidal necklaces.

1, 2, 2, 54, 4050, 1342170, 1908852102, 11488774559598, 288230375950387200, 29850020237398244599296, 12676506002282260237970435130, 21970710674130840874443091905460038, 154866286100907105149455216472736043777350, 4427744605404865645682169434028029029963535277450

Offset

0,2

Comments

The 1-dimensional (Lyndon word) case is A001037.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

Inequivalent representatives of the a(2) = 2 aperiodic necklaces:
  [0 0] [0 1]
  [0 1] [1 1]
Inequivalent representatives of the a(3) = 54 aperiodic necklaces:
  000  000  000  000  000  000  000  000  000
  000  000  001  001  001  001  001  001  001
  001  011  001  010  011  100  101  110  111
.
  000  000  000  000  000  000  000  000  000
  011  011  011  011  011  011  011  111  111
  001  010  011  100  101  110  111  001  011
.
  001  001  001  001  001  001  001  001  001
  001  001  001  001  001  001  010  010  010
  010  011  100  101  110  111  011  101  110
.
  001  001  001  001  001  001  001  001  001
  010  011  011  011  011  011  100  100  100
  111  010  011  101  110  111  011  110  111
.
  001  001  001  001  001  001  001  001  001
  101  101  101  101  110  110  110  110  111
  011  101  110  111  011  101  110  111  011
.
  001  001  001  011  011  011  011  011  011
  111  111  111  011  011  011  101  110  111
  101  110  111  101  110  111  111  111  111

Mathematica

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}]];
Table[Length[Select[(Partition[#, n]&)/@Tuples[{0, 1}, n^2], And[apermatQ[#], neckmatQ[#]]&]], {n, 4}]

Crossrefs

Main diagonal of A323861.

Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351.

Cf. A323859, A323860, A323865, A323866, A323871.

Sequence in context: A322106 A321693 A268536 * A037292 A239151 A368867

Adjacent sequences: A323869 A323870 A323871 * A323873 A323874 A323875

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Extensions

Terms a(5) and beyond from Andrew Howroyd, Aug 21 2019

Status

approved