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

2, 2, 2, 6, 8, 6, 12, 54, 54, 12, 30, 216, 486, 216, 30, 54, 990, 4020, 4020, 990, 54, 126, 3912, 32730, 64800, 32730, 3912, 126, 240, 16254, 261414, 1047540, 1047540, 261414, 16254, 240, 504, 64800, 2097018, 16764840, 33554250, 16764840, 2097018, 64800, 504

Offset: 1

Author

Gus Wiseman, Feb 04 2019

Keywords

• nonn
• tabl

Comments

The 1-dimensional case is A027375.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

Examples

			Table begins:
       1     2     3     4
    ------------------------
  1: |  2     2     6    12
  2: |  2     8    54   216
  3: |  6    54   486  4020
  4: | 12   216  4020 64800
The A(2,2) = 8 arrays:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

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

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

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

Programs

GAP

# See A323861 for code.
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

Mathematica

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

Formula

T(n,k) = n*k*A323861(n,k). - Andrew Howroyd, Aug 21 2019

Extensions

Terms a(29) and beyond from Andrew Howroyd, Aug 21 2019