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%I #7 Feb 03 2021 07:44:35
%S 1,1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,4,1,1,2,
%T 2,2,3,1,2,2,2,1,4,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,5,1,2,2,1,2,4,1,
%U 2,2,4,1,3,1,2,2,2,2,4,1,2,1,2,1,5,2,2,2
%N Number of semistandard Young tableaux whose entries 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 are weakly increasing and all columns are strictly increasing.
%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.
%C Is this a duplicate of A339887? - _R. J. Mathar_, Feb 03 2021
%F Sum_{A056239(n) = k} a(k) = A003293(n).
%e The a(60) = 5 tableaux:
%e 1123
%e .
%e 11 112 113
%e 23 3 2
%e .
%e 11
%e 2
%e 3
%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[primeMS,Join@@Permutations/@facs[n],{2}]];
%t Table[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]
%Y Cf. A000085, A000219, A003293, A053529, A056239, A112798, A138178, A153452, A296188.
%Y Cf. A323300, A323429, A323432, A323436, A323438, A323439.
%K nonn
%O 0,7
%A _Gus Wiseman_, Jan 15 2019
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