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A323869
Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.
10
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved