A176129 Number A(n,k) of solid standard Young tableaux of shape [[n*k,n],[n]]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 6, 16, 0, 1, 12, 174, 192, 0, 1, 20, 690, 7020, 2816, 0, 1, 30, 1876, 52808, 325590, 46592, 0, 1, 42, 4140, 229680, 4558410, 16290708, 835584, 0, 1, 56, 7986, 738192, 31497284, 420421056, 854630476, 15876096, 0

Offset: 0

Author

Alois P. Heinz, Jul 29 2012

Keywords

• nonn
• tabl

Comments

In general, column k is (for k > 1) asymptotic to sqrt((k+2)*(k^2 - 20*k - 8 + sqrt(k*(k+8)^3)) / (8*k^3)) * ((k+2)^(k+2)/k^k)^n / (Pi*n). - Vaclav Kotesovec, Aug 31 2014

Examples

			Square array A(n,k) begins:
  1,      1,        1,         1,          1,           1, ...
  0,      2,        6,        12,         20,          30, ...
  0,     16,      174,       690,       1876,        4140, ...
  0,    192,     7020,     52808,     229680,      738192, ...
  0,   2816,   325590,   4558410,   31497284,   146955276, ...
  0,  46592, 16290708, 420421056, 4600393936, 31113230148, ...
		

Links

• Alois P. Heinz, Antidiagonals 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=0-10 give: A000007, A006335, A214801, A215686, A246619, A246620, A246621, A246632, A246633, A246634, A246635.

Rows n=0-3 give: A000012, A002378, A215687, A215688.

Main diagonal gives: A215123.

Programs

Maple

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y), `if`(z>x, 0,
      `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:
A:= (n, k)-> b(n*k, n, n):
seq(seq(A(n, d-n), n=0..d), d=0..8);

Mathematica

b [x_, y_, z_] := b[x, y, z] = If[z > y, b[x, z, y], If[z > x, 0, 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]]]]; a[n_, k_] := b[n*k, n, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)