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

1,1

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.

Links

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

Formula

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

Example

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]

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}]

Prog

(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

Crossrefs

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

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

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

Sequence in context: A151694 A361424 A298745 * A121698 A087482 A137227

Adjacent sequences: A323857 A323858 A323859 * A323861 A323862 A323863

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 04 2019

Extensions

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

Status

approved