OFFSET
0,4
COMMENTS
The 1-dimensional (Lyndon word) case is A059966.
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..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
EXAMPLE
Inequivalent representatives of the a(6) = 18 toroidal necklaces:
[6] [1 5] [2 4] [1 1 4] [1 2 3] [1 3 2] [1 1 1 3] [1 1 2 2] [1 1 1 1 2]
.
[1] [2] [1 1]
[5] [4] [1 3]
.
[1] [1] [1]
[1] [2] [3]
[4] [3] [2]
.
[1] [1]
[1] [1]
[1] [2]
[3] [2]
.
[1]
[1]
[1]
[1]
[2]
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), SameQ@@Length/@#&];
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[If[n==0, 1, Length[Union@@Table[Select[ptnmats[k], And[apermatQ[#], neckmatQ[#]]&], {k, Times@@Prime/@#&/@IntegerPartitions[n]}]]], {n, 0, 10}]
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved