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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(5) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved