A215122 Number T(n,k) of solid standard Young tableaux of shape [[(n-k)*k,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 0, 0, 2, 0, 0, 8, 8, 0, 0, 30, 174, 30, 0, 0, 112, 2084, 2084, 112, 0, 0, 420, 21025, 52808, 21025, 420, 0, 0, 1584, 194064, 994788, 994788, 194064, 1584, 0, 0, 6006, 1694224, 16074586, 31497284, 16074586, 1694224, 6006, 0

Offset: 0

Author

Alois P. Heinz, Aug 03 2012

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,    0;
  0,    2,      0;
  0,    8,      8,      0;
  0,   30,    174,     30,      0;
  0,  112,   2084,   2084,    112,      0;
  0,  420,  21025,  52808,  21025,    420,    0;
  0, 1584, 194064, 994788, 994788, 194064, 1584,  0;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
• Wikipedia, Young tableau

Crossrefs

Columns k=0-2 give: A000007, A162551(n-1), A215124.

Central elements of rows give A215123.

Programs

Maple

b:= proc(x, y, z) option remember; `if`(zy and x>z, b(x-1, y, z), 0)+
      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
    end:
T:= (n, k)-> `if`(k=0 xor k=n, 0, b((n-k)*k, k, n-k)):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[x_, y_, z_] := b[x, y, z] = If[zy && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; T[n_, k_] := If[k == 0 || k == n, 0, b[(n-k)*k, k, n-k]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)