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

1,5

Comments

Row sums equal A003293.

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

Links

Table of n, a(n) for n=1..136.

Example

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;

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}]

Crossrefs

Cf. A003293, A005986.

Sequence in context: A092905 A052509 A172119 * A227588 A093628 A186807

Adjacent sequences: A228122 A228123 A228124 * A228126 A228127 A228128

Keyword

nonn,tabl

Author

Wouter Meeussen, Aug 11 2013

Status

approved