A323437 - OEIS

%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