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%I #19 Aug 21 2020 01:46:49
%S 1,1,3,5,11,16,33,47,85,126,208,299,486,685,1050,1496,2221,3097,4523,
%T 6239,8901,12219,17093,23202,32120,43200,58899,78761,106210,140786,
%U 188192,247689,327965,429183,563592,732730,955851,1235370,1600205,2057743,2649254
%N Number of generalized Young tableaux with constant rows, weakly increasing columns, and entries summing to n.
%C For strictly increasing columns, see A100883.
%e The a(5) = 16 tableaux:
%e 5 1 1 1 1 1
%e .
%e 1 2 1 1 1 1 1 1 1 1 1 1 1 1
%e 4 3 3 2 1 1 1
%e .
%e 1 1 1 1 1 1 1 1 1
%e 1 2 1 1 1 1
%e 3 2 2 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 comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
%t Table[Sum[Length[Select[comps[ptn],And@@SameQ@@@#&&GreaterEqual@@Length/@#&]],{ptn,Sort/@IntegerPartitions[n]}],{n,10}]
%Y Cf. A000085, A000219, A003293, A006951, A100883, A138178, A279784, A299968, A323432, A323436, A323437, A323438, A323450.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 19 2019
%E a(21)-a(40) from _Seiichi Manyama_, Aug 20 2020
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