OFFSET
1,1
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 = 1..1275
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
Andrew Howroyd, GAP Program Code
EXAMPLE
Table begins:
1 2 3 4
------------------------
1: | 2 1 2 3
2: | 1 2 9 27
3: | 2 9 54 335
4: | 3 27 335 4050
Inequivalent representatives of the A(3,2) = 9 aperiodic toroidal necklaces:
[0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1]
[0 0 1] [0 1 1] [0 1 0] [0 1 1] [1 0 1] [1 1 0] [1 1 1] [1 0 1] [1 1 1]
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-k]&/@Tuples[{0, 1}, (n-k)*k], And[apermatQ[#], neckmatQ[#]]&]], {n, 6}, {k, n-1}]
PROG
(GAP) # See link for code.
for n in [1..8] do for k in [1..8] do Print(A323861(n, k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(37) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved