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A323869
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Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.
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10
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1, 4, 24, 212, 1080, 18672, 94584, 2182752, 21261708, 408988080, 3245265144, 168549358368, 1053716696760, 42565371692592, 921132763909200, 26578273403903040, 260741534058271800, 20313207979498492344, 185603174638656822264, 16066126777465282744800, 324499299994016295338064
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OFFSET
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1,2
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COMMENTS
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An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.
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LINKS
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FORMULA
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EXAMPLE
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The a(3) = 24 matrices:
[123][132][213][312][231][321][122][211][112][221][121][212]
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[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]
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MATHEMATICA
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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}]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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