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A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.

%I A228125 #21 Jul 31 2016 09:03:15
%S A228125 1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,7,5,2,1,1,6,10,9,5,2,1,1,7,14,16,
%T A228125 10,5,2,1,1,8,19,24,19,11,5,2,1,1,9,24,37,32,21,11,5,2,1,1,10,30,51,
%U A228125 52,38,22,11,5,2,1,1,11,37,71,79,66,41,23,11,5,2,1,1,12,44,93,117,106,74,43,23,11,5,2,1,1,13,52,122,166,166,125,80,44,23,11,5,2,1,1,14,61,153,231,251,204,139,83,45,23,11,5,2,1,1,15,70,193,311,367,322,236,147,85,45,23,11,5,2,1
%N A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.
%C A228125 Row sums equal A003293.
%C A228125 Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ...
%e A228125 T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are
%e A228125 114 , 123 , 222 , 11 ,  12  , 13 ,   1
%e A228125                   4     3     2      2
%e A228125                                      3
%e A228125 Triangle starts:
%e A228125 1;
%e A228125 1,  1;
%e A228125 1,  2,  1;
%e A228125 1,  3,  2,  1;
%e A228125 1,  4,  4,  2,  1;
%e A228125 1,  5,  7,  5,  2,  1;
%e A228125 1,  6, 10,  9,  5,  2,  1;
%e A228125 1,  7, 14, 16, 10,  5,  2,  1;
%e A228125 1,  8, 19, 24, 19, 11,  5,  2, 1;
%e A228125 1,  9, 24, 37, 32, 21, 11,  5, 2, 1;
%e A228125 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1;
%t A228125 hooklength[(par_)?PartitionQ]:=Table[Count[par,q_ /; q>=j] +1-i +par[[i]] -j, {i,Length[par]}, {j,par[[i]]} ];
%t A228125 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 A228125 Cf. A003293, A005986.
%K A228125 nonn,tabl
%O A228125 1,5
%A A228125 _Wouter Meeussen_, Aug 11 2013