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A323867 revision #2

A323867

Number of aperiodic arrays of positive integers summing to n.

1, 1, 1, 5, 11, 33, 57, 157, 303, 683, 1358, 2974, 5932, 12560, 25328, 52400

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OFFSET

0,4

COMMENTS

The 1-dimensional case is A000740.

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

Table of n, a(n) for n=0..15.

EXAMPLE

The a(5) = 33 arrays:

5 14 23 32 41 113 122 131 212 221 311 1112 1121 1211 2111

1 2 3 4 11 11 12 21

4 3 2 1 12 21 11 11

1 1 1 2 2 3

1 2 3 1 2 1

3 2 1 2 1 1

1 1 1 2

1 1 2 1

1 2 1 1

2 1 1 1

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), SameQ@@Length/@#&];

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];

Table[Length[Union@@Table[Select[ptnmats[k], apermatQ], {k, Times@@Prime/@#&/@IntegerPartitions[n]}]], {n, 15}]

CROSSREFS

Cf. A000670, A000740, A027375, A101509.

Cf. A323860, A323861, A323862, A323863, A323864, A323866, A323869.

Sequence in context: A106908 A107442 A349611 * * A280540 A077917

Adjacent sequences: A323864 A323865 A323866 * A323868 A323869 A323870

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Feb 04 2019

STATUS

editing