A259479 Skew diagrams, both connected or not.

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 5, 2, 0, 0, 0, 11, 9, 6, 1, 0, 0, 0, 15, 13, 12, 6, 0, 0, 0, 0, 22, 20, 22, 14, 3, 0, 0, 0, 0, 30, 28, 36, 27, 13, 2, 0, 0, 0, 0, 42, 40, 56, 48, 31, 11, 1, 0, 0, 0, 0, 56, 54, 82, 77, 59, 33, 9, 0, 0, 0, 0, 0, 77, 75, 120, 121, 106, 72, 30, 6, 0, 0, 0, 0, 0

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.

Links

Table of n, a(n) for n=0..90.

Wouter Meeussen, Table n, m, T(n,m) for n= 1..27

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

Cf. A259478, A259480, A259481, A161492, A227309, A006958.

Sequence in context: A287736 A180969 A369312 * A238343 A238128 A238121

Adjacent sequences: A259476 A259477 A259478 * A259480 A259481 A259482

Keyword

nonn,tabl

Author

Wouter Meeussen, Jun 28 2015

Status

approved