A323860 - OEIS

%I #9 Aug 22 2019 20:47:18

%S 2,2,2,6,8,6,12,54,54,12,30,216,486,216,30,54,990,4020,4020,990,54,

%T 126,3912,32730,64800,32730,3912,126,240,16254,261414,1047540,1047540,

%U 261414,16254,240,504,64800,2097018,16764840,33554250,16764840,2097018,64800,504

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

%C The 1-dimensional case is A027375.

%C 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 Andrew Howroyd, <a href="/A323860/b323860.txt">Table of n, a(n) for n = 1..1275</a>

%F T(n,k) = n*k*A323861(n,k). - _Andrew Howroyd_, Aug 21 2019

%e Table begins:

%e        1     2     3     4

%e     ------------------------

%e   1: |  2     2     6    12

%e   2: |  2     8    54   216

%e   3: |  6    54   486  4020

%e   4: | 12   216  4020 64800

%e The A(2,2) = 8 arrays:

%e   [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]

%e   [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]

%e Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:

%e   [1 0] [0 1]

%e   [0 1] [1 0]

%t apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];

%t Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

%o (GAP) # See A323861 for code.

%o 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 First and last columns are A027375. Main diagonal is A323863.

%Y Cf. A000740, A001037, A179043, A265627, A323351.

%Y Cf. A323861, A323862, A323864, A323865, A323867, A323869.

%K nonn,tabl

%O 1,1

%A _Gus Wiseman_, Feb 04 2019

%E Terms a(29) and beyond from _Andrew Howroyd_, Aug 21 2019