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

Author

Wouter Meeussen, Aug 11 2013

Keywords

• nonn
• tabf

Comments

Row sums are A228129.

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

Examples

			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".
		

Links

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

Crossrefs

Cf. A228125, A228128, A228129.

Programs

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}]