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.

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, 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, 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

Offset: 1

Author

Wouter Meeussen, Aug 11 2013

Keywords

• nonn
• tabl

Comments

Row sums equal A003293.

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

Examples

			T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are
114 , 123 , 222 , 11 ,  12  , 13 ,   1
                  4     3     2      2
                                     3
Triangle starts:
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, 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, 52, 38, 22, 11, 5, 2, 1;
		

Crossrefs

Cf. A003293, A005986.

Programs

Mathematica

hooklength[(par_)?PartitionQ]:=Table[Count[par,q_ /; q>=j] +1-i +par[[i]] -j, {i,Length[par]}, {j,par[[i]]} ];
Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1&) /@ Flatten[hooklength[#1]],{q,0,w}]&) /@ Partitions[n]],{w,24},{n,w}]