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A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

%I A323860 #9 Aug 22 2019 20:47:18
%S A323860 2,2,2,6,8,6,12,54,54,12,30,216,486,216,30,54,990,4020,4020,990,54,
%T A323860 126,3912,32730,64800,32730,3912,126,240,16254,261414,1047540,1047540,
%U A323860 261414,16254,240,504,64800,2097018,16764840,33554250,16764840,2097018,64800,504
%N A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.
%C A323860 The 1-dimensional case is A027375.
%C A323860 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 A323860 Andrew Howroyd, <a href="/A323860/b323860.txt">Table of n, a(n) for n = 1..1275</a>
%F A323860 T(n,k) = n*k*A323861(n,k). - _Andrew Howroyd_, Aug 21 2019
%e A323860 Table begins:
%e A323860        1     2     3     4
%e A323860     ------------------------
%e A323860   1: |  2     2     6    12
%e A323860   2: |  2     8    54   216
%e A323860   3: |  6    54   486  4020
%e A323860   4: | 12   216  4020 64800
%e A323860 The A(2,2) = 8 arrays:
%e A323860   [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
%e A323860   [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
%e A323860 Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
%e A323860   [1 0] [0 1]
%e A323860   [0 1] [1 0]
%t A323860 apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
%t A323860 Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]
%o A323860 (GAP) # See A323861 for code.
%o A323860 for n in [1..8] do for k in [1..8] do Print(n*k*A323861(n,k), ", "); od; Print("\n"); od; # _Andrew Howroyd_, Aug 21 2019
%Y A323860 First and last columns are A027375. Main diagonal is A323863.
%Y A323860 Cf. A000740, A001037, A179043, A265627, A323351.
%Y A323860 Cf. A323861, A323862, A323864, A323865, A323867, A323869.
%K A323860 nonn,tabl
%O A323860 1,1
%A A323860 _Gus Wiseman_, Feb 04 2019
%E A323860 Terms a(29) and beyond from _Andrew Howroyd_, Aug 21 2019