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A323439 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.

%I A323439 #5 Jan 17 2019 09:57:53
%S A323439 1,1,1,0,1,2,1,0,0,2,1,0,1,2,2,0,1,1,1,0,2,2,1,0,0,2,0,0,1,4,1,0,2,2,
%T A323439 2,0,1,2,2,0,1,4,1,0,0,2,1,0,0,1,2,0,1,0,2,0,2,2,1,0,1,2,0,0,2,4,1,0,
%U A323439 2,4,1,0,1,2,1,0,2,4,1,0,0,2,1,0,2,2,2
%N A323439 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.
%C A323439 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.
%H A323439 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A323439 Sum_{A056239(n) = k} a(k) = A323451(n).
%e A323439 The a(630) = 8 tableaux:
%e A323439   123   124   1234
%e A323439   24    23    2
%e A323439 .
%e A323439   12   12   123   124
%e A323439   23   24   2     2
%e A323439   4    3    4     3
%e A323439 .
%e A323439   12
%e A323439   2
%e A323439   3
%e A323439   4
%t A323439 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323439 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323439 ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
%t A323439 Table[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(Less@@@DeleteCases[Transpose[PadRight[#]],0,{2}]),And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]
%Y A323439 Cf. A000085, A000219, A056239, A112798, A114736, A117433, A299968, A323300, A323430, A323436, A323438, A323450, A323451.
%K A323439 nonn
%O A323439 1,6
%A A323439 _Gus Wiseman_, Jan 16 2019