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A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.

%I A323451 #20 Aug 19 2020 10:09:15
%S A323451 1,1,1,3,3,6,9,12,19,27,39,54,79,107,150,209,282,387,525,707,949,1272,
%T A323451 1688,2244,2968,3902,5125,6712,8752,11383,14780,19109,24671,31768,
%U A323451 40791,52280,66860,85296,108621,138054,175085,221676,280161,353414,445098,559661
%N A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.
%C A323451 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 A323451 Seiichi Manyama, <a href="/A323451/b323451.txt">Table of n, a(n) for n = 0..50</a>
%H A323451 nLab, <a href="https://ncatlab.org/nlab/show/Young+diagram">Young Diagram</a>.
%H A323451 The Unapologetic Mathematician weblog, <a href="https://unapologetic.wordpress.com/2011/02/02/generalized-young-tableaux/">Generalized Young Tableaux</a>.
%e A323451 The a(8) = 19 generalized Young tableaux:
%e A323451   8   1 7   2 6   3 5   1 2 5   1 3 4
%e A323451 .
%e A323451   1   2   3   1 2   1 5   1 3   1 4   2 3   1 2   1 2 3
%e A323451   7   6   5   5     2     4     3     3     2 3   2
%e A323451 .
%e A323451   1   1   1 2
%e A323451   2   3   2
%e A323451   5   4   3
%t A323451 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323451 sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A323451 ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@sqfacs[n],{2}]];
%t A323451 Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&&And@@(UnsameQ@@@DeleteCases[Transpose[PadRight[#]],0,{2}])&]],{y,IntegerPartitions[n]}],{n,10}]
%Y A323451 Cf. A000085, A000219, A003293, A114736, A117433, A138178, A299968.
%Y A323451 Cf. A323430, A323436, A323437, A323439, A323450.
%K A323451 nonn
%O A323451 0,4
%A A323451 _Gus Wiseman_, Jan 16 2019
%E A323451 a(21)-a(45) from _Seiichi Manyama_, Aug 19 2020