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A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

%I A323438 #5 Jan 17 2019 09:57:46
%S A323438 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,3,1,4,2,2,1,7,2,2,3,4,1,4,1,7,2,2,
%T A323438 2,8,1,2,2,7,1,4,1,4,4,2,1,12,2,3,2,4,1,5,2,7,2,2,1,10,1,2,4,11,2,4,1,
%U A323438 4,2,4,1,13,1,2,3,4,2,4,1,12,5,2,1,10,2
%N A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.
%C A323438 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 A323438 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A323438 Sum_{A056239(n) = k} a(k) = A323450(n).
%e A323438 The a(96) = 19 tableaux:
%e A323438   111112
%e A323438 .
%e A323438   111   1111   1112   11111   11112
%e A323438   112   12     11     2       1
%e A323438 .
%e A323438   11   111   111   112   1111   1112
%e A323438   11   11    12    11    1      1
%e A323438   12   2     1     1     2      1
%e A323438 .
%e A323438   11   11   111   112
%e A323438   11   12   1     1
%e A323438   1    1    1     1
%e A323438   2    1    2     1
%e A323438 .
%e A323438   11   12
%e A323438   1    1
%e A323438   1    1
%e A323438   1    1
%e A323438   2    1
%e A323438 .
%e A323438   1
%e A323438   1
%e A323438   1
%e A323438   1
%e A323438   1
%e A323438   2
%t A323438 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323438 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323438 ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
%t A323438 Table[Length[Select[ptnplane[y],And[And@@LessEqual@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]
%Y A323438 Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A299968, A323300, A323430, A323436, A323438, A323450.
%K A323438 nonn
%O A323438 1,4
%A A323438 _Gus Wiseman_, Jan 16 2019