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A323861
Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.
14
2, 1, 1, 2, 2, 2, 3, 9, 9, 3, 6, 27, 54, 27, 6, 9, 99, 335, 335, 99, 9, 18, 326, 2182, 4050, 2182, 326, 18, 30, 1161, 14523, 52377, 52377, 14523, 1161, 30, 56, 4050, 99858, 698535, 1342170, 698535, 99858, 4050, 56, 99, 14532, 698870, 9586395, 35790267, 35790267, 9586395, 698870, 14532, 99
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
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
First and last columns are A001037. Main diagonal is A323872.
Sequence in context: A131059 A133098 A145788 * A117592 A117942 A290092
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(37) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved