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A228128 T(n,m) = semistandard Young tableau families, headed by a father SSYT with shape a partition of k, containing daughter SSYT of shape equal to once-trimmed father's shape, so that union of families equals all SSYT with sum of entries n.

%I A228128 #16 Aug 13 2013 23:36:37
%S A228128 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,3,3,1,1,0,1,3,4,3,1,1,0,1,4,7,5,3,
%T A228128 1,1,0,1,5,8,9,6,3,1,1,0,1,5,13,13,10,6,3,1,1,0,1,6,14,20,17,11,6,3,1,
%U A228128 1,0,1,7,20,27,28,19,12,6,3,1,1,0,1,7,22,38,40,33,20,12,6,3,1,1,0,1,8,29,49,60,51,37,21,12,6,3,1,1,0,1,9,31,65,85,79,59,39,22,12,6,3,1,1
%N A228128 T(n,m) = semistandard Young tableau families, headed by a father SSYT with shape a partition of k, containing daughter SSYT of shape equal to once-trimmed father's shape, so that union of families equals all SSYT with sum of entries n.
%C A228128 Row sums are A228129.
%C A228128 Reverse of rows seem to converge to first differences of A005986.
%H A228128 N. Dragon, R. Stanley, <a href="http://mathoverflow.net/questions/129854">Semi-Standard Young Diagrams and families</a>;
%H A228128 N. Dragon, <a href="http://www.itp.uni-hannover.de/~dragon/young.pdf"> résumé</a>
%e A228128 T(6,3) = 3 since the 7 tableaux in the family contain 3 father tableaux:
%e A228128 11  ,  13  ,  1
%e A228128 4      2      2
%e A228128               3
%e A228128 see 2nd link, "content 6".
%t A228128 (* hooklength: see A228125 *);
%t A228128 Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1 &) /@ Flatten[hooklength[#1]],{q,0,w}]& ) /@ Partitions[n]],{w,24},{n,w}]
%Y A228128 Cf. A228125, A228128, A228129.
%K A228128 nonn,tabf
%O A228128 1,13
%A A228128 _Wouter Meeussen_, Aug 11 2013