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%I #16 Aug 19 2020 10:07:49
%S 1,1,3,6,14,26,56,103,203,374,702,1262,2306,4078,7242,12628,21988,
%T 37756,64682,109606,185082,309958,516932,856221,1412461,2316416,
%U 3783552
%N Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.
%C A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.
%H nLab, <a href="https://ncatlab.org/nlab/show/Young+diagram">Young Diagram</a>.
%H The Unapologetic Mathematician weblog, <a href="https://unapologetic.wordpress.com/2011/02/02/generalized-young-tableaux/">Generalized Young Tableaux</a>.
%e The a(4) = 14 generalized Young tableaux:
%e 4 1 3 2 2 1 1 2 1 1 1 1
%e .
%e 1 2 1 1 1 2 1 1 1 1 1
%e 3 2 2 1 1 1 1
%e .
%e 1 1 1
%e 1 1
%e 2 1
%e .
%e 1
%e 1
%e 1
%e 1
%e The a(5) = 26 generalized Young tableaux:
%e 5 1 4 2 3 1 1 3 1 2 2 1 1 1 2 1 1 1 1 1
%e .
%e 1 2 1 1 1 3 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1
%e 4 3 3 1 2 1 2 2 1 1 1 1
%e .
%e 1 1 1 1 1 2 1 1 1 1 1
%e 1 2 1 1 1 1 1
%e 3 2 2 1 1 1
%e .
%e 1 1 1
%e 1 1
%e 1 1
%e 2 1
%e .
%e 1
%e 1
%e 1
%e 1
%e 1
%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[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&]],{y,IntegerPartitions[n]}],{n,10}]
%Y Cf. A000085, A000219, A003293, A053529, A114736, A138178, A296188, A299968.
%Y Cf. A323436, A323437, A323438, A323439, A323451.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jan 16 2019
%E a(16)-a(26) from _Seiichi Manyama_, Aug 19 2020
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