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A323585
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Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.
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4
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1, 1, 0, 3, 7, 21, 30, 83, 129, 267, 428, 856, 1332, 2482, 3909, 6798, 10853, 18331, 28665, 47327, 73829, 118527, 183898, 290780, 446508, 695964, 1061290, 1631829, 2470970, 3759609, 5646952, 8512306, 12700005, 18972387, 28120953, 41690725, 61392966, 90379781
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OFFSET
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0,4
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COMMENTS
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A multiset is aperiodic if its multiplicities are relatively prime.
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LINKS
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FORMULA
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The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the second is A323584.
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EXAMPLE
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The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:
31 211
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3 21 111
1 1 1
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2 11
1 1
1 1
The same for a(5) = 21:
41 32 311 221 2111
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4 3 31 21 22 21 211 111 1111
1 2 1 2 1 11 1 11 1
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3 2 21 11 111
1 2 1 11 1
1 1 1 1 1
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2 11
1 1
1 1
1 1
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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]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS, Join@@Permutations/@facs[n], {2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And[GCD@@Length/@Split[#]==1, GCD@@Length/@Split[Transpose[PadRight[#]]]==1, And@@GreaterEqual@@@#, And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]], {y, Select[IntegerPartitions[n], GCD@@#==1&]}], {n, 10}]
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CROSSREFS
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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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