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.

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, 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, 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

Offset

1,13

Comments

Row sums are A228129.

Reverse of rows seem to converge to first differences of A005986.

Links

Table of n, a(n) for n=1..120.

N. Dragon, R. Stanley, Semi-Standard Young Diagrams and families;

N. Dragon, résumé

Example

T(6,3) = 3 since the 7 tableaux in the family contain 3 father tableaux:
11  ,  13  ,  1
4      2      2
              3
see 2nd link, "content 6".

Mathematica

(* hooklength: see 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}]

Crossrefs

Cf. A228125, A228128, A228129.

Sequence in context: A191607 A029387 A070878 * A060959 A342689 A077042

Adjacent sequences: A228125 A228126 A228127 * A228129 A228130 A228131

Keyword

nonn,tabf

Author

Wouter Meeussen, Aug 11 2013

Status

approved