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A323302
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Number of ways to arrange the parts of the integer partition with Heinz number n into a matrix with equal row-sums and equal column-sums.
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6
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1, 1, 1, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 0, 3, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0
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OFFSET
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1,4
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COMMENTS
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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
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LINKS
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EXAMPLE
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The a(900) = 12 matrix-arrangements of (3,3,2,2,1,1):
[1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
[3 2 1] [3 1 2] [2 3 1] [2 1 3] [1 3 2] [1 2 3]
.
[1 3] [1 3] [2 2] [2 2] [3 1] [3 1]
[2 2] [3 1] [1 3] [3 1] [1 3] [2 2]
[3 1] [2 2] [3 1] [1 3] [2 2] [1 3]
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MATHEMATICA
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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/@#&];
Table[Length[Select[ptnmats[n], And[SameQ@@Total/@#, SameQ@@Total/@Transpose[#]]&]], {n, 100}]
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CROSSREFS
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Positions of zeros are a superset of A106543.
Cf. A000005, A001222, A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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