A259480 T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n).

0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 0, 0, 0, 7, 2, 0, 0, 0, 0, 11, 5, 2, 0, 0, 0, 0, 15, 8, 4, 0, 0, 0, 0, 0, 22, 14, 10, 3, 0, 0, 0, 0, 0, 30, 21, 18, 7, 1, 0, 0, 0, 0, 0, 42, 32, 32, 17, 6, 0, 0, 0, 0, 0, 0, 56, 45, 50, 31, 15, 2, 0, 0, 0, 0, 0, 0, 77, 65, 80, 58, 36, 11, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Wouter Meeussen, Jul 01 2015

Keywords

• nonn
• tabl

Comments

In contrast to A161492, which counts the same items by area and number of columns, this sequence appears to have no known generating function.

The diagonals T(n,n-k) count connected skew diagrams with weight k:

1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;

Their sums equal A006958.

Examples

			T(7,2) = 4, the pairs of partitions are ((4,3)/(2)), ((3,3,1)/(2)), ((3,2,2)/(1,1)) and ((2,2,2,1)/(1,1));
The diagrams are:
  x x 0 0 , x x 0 , x 0 0 , x 0
  0 0 0     0 0 0   x 0     x 0
            0       0 0     0 0
                            0
Triangle begins:
      k=0  1  2  3  4  5  6  7
  n=0;  0
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  0  0  0
  n=4;  5  1  0  0  0
  n=5;  7  2  0  0  0  0
  n=6; 11  5  2  0  0  0  0
  n=7; 15  8  4  0  0  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, A259479, A259481, A161492, A227309, A006958.

Programs

Mathematica

(* see A259479 *) factor[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=Block[{a1,a2,a3},a1=Apply[Join,Table[{i,j},{i,Length[\[Lambda]]},{j,\[Lambda][[i]],\[Lambda][[Min[i+1,Length[\[Lambda]]]]],-1}]];
a2=Map[{First[#],First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&,a1];a3=Map[First,DeleteCases[SplitBy[a2,MatchQ[#,{,False}]&],{{,False}}],{2}];
Flatten[redu[Part[\[Lambda],#], Part[PadRight[\[Mu],Length[\[Lambda]],0],#]/. 0->Sequence[]]&/@Map[Union,a3],1]];
Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&redu[\[Lambda],\[Mu]]==factor[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}]