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

Author

Wouter Meeussen, Jun 28 2015

Keywords

• nonn
• tabl

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.

Examples

			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
		

References

• I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.

Links

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

Crossrefs

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

Programs

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}];