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%I #4 Jan 20 2019 08:56:20
%S 1,1,1,3,3,5,8,10,14,19,28,34,48,60,80,106,134,171,222,279,354,452,
%T 562,706,884,1100
%N Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.
%H nLab, <a href="https://ncatlab.org/nlab/show/Young+diagram">Young Diagram</a>.
%e The a(8) = 14 tableaux:
%e 8 1 7 2 6 3 5 1 2 5 1 3 4
%e .
%e 7 6 5 2 5 3 4 2 3
%e 1 2 3 1 1 1 2
%e .
%e 5 4
%e 2 3
%e 1 1
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,sqfacs[y],{2}],And@@Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}]&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]
%Y Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323580.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Jan 18 2019
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