A323436 - OEIS

%I #4 Jan 15 2019 18:45:43

%S 1,1,1,1,2,1,2,1,3,2,2,1,3,1,2,2,5,1,4,1,3,2,2,1,5,2,2,3,3,1,4,1,7,2,

%T 2,2,8,1,2,2,5,1,4,1,3,3,2,1,7,2,4,2,3,1,7,2,5,2,2,1,8,1,2,3,11,2,4,1,

%U 3,2,4,1,12,1,2,4,3,2,4,1,7,5,2,1,8,2,2

%N Number of plane partitions whose parts are the prime indices of n.

%C Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly decreasing.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%e The a(120) = 12 plane partitions:

%e   32111

%e .

%e   311   321   3111   3211

%e   21    11    2      1

%e .

%e   31   32   311   321

%e   21   11   2     1

%e   1    1    1     1

%e .

%e   31   32

%e   2    1

%e   1    1

%e   1    1

%e .

%e   3

%e   2

%e   1

%e   1

%e   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[Length[Select[ptnplane[y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,100}]

%Y Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A296188, A299968.

%Y Cf. A323300, A323429, A323437, A323438, A323439.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jan 15 2019