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

0,5

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

Example

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;

Maple

b:= proc(x, y, z) option remember; `if`(z<y, b(x, z, y),
      `if`({x, y, z}={0}, 1, `if`(x>y 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[z<y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && 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 *)

Crossrefs

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

Central elements of rows give A215123.

Sequence in context: A058347 A058547 A230910 * A332473 A235789 A236925

Adjacent sequences: A215119 A215120 A215121 * A215123 A215124 A215125

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 03 2012

Status

approved