OFFSET
0,4
COMMENTS
T(n,m) counts pairs of partitions lambda of n and mu of 0<=m<=n respectively, so that the Ferrers diagram of mu does not exceed that of lambda, and that the diagrams of lambda and mu do not contain equal rows or columns.
REFERENCES
I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.
EXAMPLE
T(6,2) = 6, the pairs of partitions are ((4,2)/(2)), ((3,3)/(2), ((3,2,1)/(2)), ((3,2,1)/(1,1)), ((2,2,2)/(1,1)) and ((2,2,1,1)/(1,1))
and the diagrams are:
x x 0 0 , x x 0 , x x 0 , x 0 0 , x 0 , x 0
0 0 0 0 0 0 0 x 0 x 0 x 0
0 0 0 0 0
0
triangle begins:
k=0; 1 2 3 4 5 6
n=0; 1
n=1; 1 0
n=2; 2 0 0
n=3; 3 1 0 0
n=4; 5 3 0 0 0
n=5; 7 5 2 0 0 0
n=6; 11 9 6 1 0 0 0
MATHEMATICA
majorsweak[left_List, right_List]:=Block[{le1=Length[left], le2=Length[right]}, If[le2>le1||Min[Sign[left-PadRight[right, le1]]]<0, False, True]];
redu1[\[Lambda]_, \[Mu]_]/; majorsweak[\[Lambda], \[Mu]]:=Delete[#, List/@DeleteCases[Table[i Boole[\[Lambda][[i]]==\[Mu][[i]]], {i, Length[\[Mu]]}], 0]]&/@{\[Lambda], \[Mu]};
redu[\[Lambda]_, \[Mu]_]/; majorsweak[\[Lambda], \[Mu]]:=TransposePartition/@Apply[redu1, TransposePartition/@redu1[\[Lambda], \[Mu]]];
Table[Sum[Boole[majorsweak[\[Lambda], \[Mu]]&&redu[\[Lambda], \[Mu]]=={\[Lambda], \[Mu]}], {\[Lambda], Partitions[n]}, {\[Mu], Partitions[k]}], {n, 0, 12}, {k, 0, n}];
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, Jun 28 2015
STATUS
proposed