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

Author

Gus Wiseman, Feb 04 2019

Keywords

• nonn

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.

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

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

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

Programs

GAP

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

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}]

Formula

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

Extensions

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