A323869 Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.

1, 4, 24, 212, 1080, 18672, 94584, 2182752, 21261708, 408988080, 3245265144, 168549358368, 1053716696760, 42565371692592, 921132763909200, 26578273403903040, 260741534058271800, 20313207979498492344, 185603174638656822264, 16066126777465282744800, 324499299994016295338064

Offset

1,2

Comments

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..200

Formula

a(n) = n*A323871(n). - Andrew Howroyd, Aug 21 2019

Example

The a(3) = 24 matrices:
  [123][132][213][312][231][321][122][211][112][221][121][212]
.
  [1][1][2][3][2][3][1][2][1][2][1][2]
  [2][3][1][1][3][2][2][1][1][2][2][1]
  [3][2][3][2][1][1][2][1][2][1][1][2]

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn, {i, j}][[1, 1]], {i, d}, {j, n/d}], {stn, Join@@Permutations/@sps[Tuples[{Range[d], Range[n/d]}]]}], {d, Divisors[n]}];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];
Table[Length[Select[nrmmats[n], apermatQ]], {n, 6}]

Prog

(GAP) List([1..30], A323869); # See A323861 for code; Andrew Howroyd, Aug 21 2019

Crossrefs

Cf. A000670, A000740, A027375, A060223, A265627.

Cf. A323860, A323864, A323867, A323868, A323871.

Sequence in context: A245407 A162314 A369723 * A112141 A077555 A166881

Adjacent sequences: A323866 A323867 A323868 * A323870 A323871 A323872

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Extensions

Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019

Status

approved