A228128 - OEIS

%I #16 Aug 13 2013 23:36:37

%S 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,

%T 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,

%U 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

%N 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.

%C Row sums are A228129.

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

%H N. Dragon, R. Stanley, <a href="http://mathoverflow.net/questions/129854">Semi-Standard Young Diagrams and families</a>;

%H N. Dragon, <a href="http://www.itp.uni-hannover.de/~dragon/young.pdf"> résumé</a>

%e T(6,3) = 3 since the 7 tableaux in the family contain 3 father tableaux:

%e 11  ,  13  ,  1

%e 4      2      2

%e               3

%e see 2nd link, "content 6".

%t (* hooklength: see A228125 *);

%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. A228125, A228128, A228129.

%K nonn,tabf

%O 1,13

%A _Wouter Meeussen_, Aug 11 2013