A323585 - OEIS

%I #7 Jan 22 2019 16:42:11

%S 1,1,0,3,7,21,30,83,129,267,428,856,1332,2482,3909,6798,10853,18331,

%T 28665,47327,73829,118527,183898,290780,446508,695964,1061290,1631829,

%U 2470970,3759609,5646952,8512306,12700005,18972387,28120953,41690725,61392966,90379781

%N 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.

%C A multiset is aperiodic if its multiplicities are relatively prime.

%H Alois P. Heinz, <a href="/A323585/b323585.txt">Table of n, a(n) for n = 0..10000</a>

%F 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.

%e The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:

%e   31   211

%e .

%e   3   21   111

%e   1   1    1

%e .

%e   2   11

%e   1   1

%e   1   1

%e The same for a(5) = 21:

%e   41   32   311   221   2111

%e .

%e   4   3   31   21   22   21   211   111   1111

%e   1   2   1    2    1    11   1     11    1

%e .

%e   3   2   21   11   111

%e   1   2   1    11   1

%e   1   1   1    1    1

%e .

%e   2   11

%e   1   1

%e   1   1

%e   1   1

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

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

%t ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];

%t 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}]

%Y Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, A323587.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 19 2019