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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A235789 A236925 A134414

Adjacent sequences:  A215119 A215120 A215121 * A215123 A215124 A215125

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 03 2012

STATUS

approved

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Last modified June 18 07:06 EDT 2019. Contains 324203 sequences. (Running on oeis4.)