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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. 5

%I

%S 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 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 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 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 Row sums equal A003293.

%C Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ...

%e T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are

%e 114 , 123 , 222 , 11 , 12 , 13 , 1

%e 4 3 2 2

%e 3

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 2, 1;

%e 1, 4, 4, 2, 1;

%e 1, 5, 7, 5, 2, 1;

%e 1, 6, 10, 9, 5, 2, 1;

%e 1, 7, 14, 16, 10, 5, 2, 1;

%e 1, 8, 19, 24, 19, 11, 5, 2, 1;

%e 1, 9, 24, 37, 32, 21, 11, 5, 2, 1;

%e 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1;

%t hooklength[(par_)?PartitionQ]:=Table[Count[par,q_ /; q>=j] +1-i +par[[i]] -j, {i,Length[par]}, {j,par[[i]]} ];

%t 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 Cf. A003293, A005986.

%K nonn,tabl

%O 1,5

%A _Wouter Meeussen_, Aug 11 2013

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Last modified July 22 16:43 EDT 2019. Contains 325225 sequences. (Running on oeis4.)