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A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

%I A323861 #14 Aug 22 2019 20:47:29
%S A323861 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,
%T A323861 2182,326,18,30,1161,14523,52377,52377,14523,1161,30,56,4050,99858,
%U A323861 698535,1342170,698535,99858,4050,56,99,14532,698870,9586395,35790267,35790267,9586395,698870,14532,99
%N A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.
%C A323861 The 1-dimensional (Lyndon word) case is A001037.
%C A323861 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.
%H A323861 Andrew Howroyd, <a href="/A323861/b323861.txt">Table of n, a(n) for n = 1..1275</a>
%H A323861 S. N. Ethier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">Counting toroidal binary arrays</a>, J. Int. Seq. 16 (2013) #13.4.7.
%H A323861 Andrew Howroyd, <a href="/A323861/a323861.txt">GAP Program Code</a>
%e A323861 Table begins:
%e A323861         1    2    3    4
%e A323861     ------------------------
%e A323861   1: |  2    1    2    3
%e A323861   2: |  1    2    9   27
%e A323861   3: |  2    9   54  335
%e A323861   4: |  3   27  335 4050
%e A323861 Inequivalent representatives of the A(3,2) = 9 aperiodic toroidal necklaces:
%e A323861   [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]
%e A323861   [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]
%t A323861 apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
%t A323861 neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
%t A323861 Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],And[apermatQ[#],neckmatQ[#]]&]],{n,6},{k,n-1}]
%o A323861 (GAP) # See link for code.
%o A323861 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
%Y A323861 First and last columns are A001037. Main diagonal is A323872.
%Y A323861 Cf. A000031, A000740, A027375, A059966, A179043, A184271.
%Y A323861 Cf. A323859, A323860, A323865, A323866, A323871.
%K A323861 nonn,tabl
%O A323861 1,1
%A A323861 _Gus Wiseman_, Feb 04 2019
%E A323861 Terms a(37) and beyond from _Andrew Howroyd_, Aug 21 2019