cp's OEIS Frontend

A323872 Number of n X n aperiodic binary toroidal necklaces.

%I A323872 #10 Aug 22 2019 22:14:39
%S A323872 1,2,2,54,4050,1342170,1908852102,11488774559598,288230375950387200,
%T A323872 29850020237398244599296,12676506002282260237970435130,
%U A323872 21970710674130840874443091905460038,154866286100907105149455216472736043777350,4427744605404865645682169434028029029963535277450
%N A323872 Number of n X n aperiodic binary toroidal necklaces.
%C A323872 The 1-dimensional (Lyndon word) case is A001037.
%C A323872 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 A323872 Andrew Howroyd, <a href="/A323872/b323872.txt">Table of n, a(n) for n = 0..50</a>
%e A323872 Inequivalent representatives of the a(2) = 2 aperiodic necklaces:
%e A323872   [0 0] [0 1]
%e A323872   [0 1] [1 1]
%e A323872 Inequivalent representatives of the a(3) = 54 aperiodic necklaces:
%e A323872   000  000  000  000  000  000  000  000  000
%e A323872   000  000  001  001  001  001  001  001  001
%e A323872   001  011  001  010  011  100  101  110  111
%e A323872 .
%e A323872   000  000  000  000  000  000  000  000  000
%e A323872   011  011  011  011  011  011  011  111  111
%e A323872   001  010  011  100  101  110  111  001  011
%e A323872 .
%e A323872   001  001  001  001  001  001  001  001  001
%e A323872   001  001  001  001  001  001  010  010  010
%e A323872   010  011  100  101  110  111  011  101  110
%e A323872 .
%e A323872   001  001  001  001  001  001  001  001  001
%e A323872   010  011  011  011  011  011  100  100  100
%e A323872   111  010  011  101  110  111  011  110  111
%e A323872 .
%e A323872   001  001  001  001  001  001  001  001  001
%e A323872   101  101  101  101  110  110  110  110  111
%e A323872   011  101  110  111  011  101  110  111  011
%e A323872 .
%e A323872   001  001  001  011  011  011  011  011  011
%e A323872   111  111  111  011  011  011  101  110  111
%e A323872   101  110  111  101  110  111  111  111  111
%t A323872 apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
%t A323872 neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
%t A323872 Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}]
%Y A323872 Main diagonal of A323861.
%Y A323872 Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351.
%Y A323872 Cf. A323859, A323860, A323865, A323866, A323871.
%K A323872 nonn
%O A323872 0,2
%A A323872 _Gus Wiseman_, Feb 04 2019
%E A323872 Terms a(5) and beyond from _Andrew Howroyd_, Aug 21 2019