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

0,5

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

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

Example

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, ...

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 *)

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.

Sequence in context: A337107 A362588 A367073 * A362787 A341200 A300130

Adjacent sequences: A176126 A176127 A176128 * A176130 A176131 A176132

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 29 2012

Status

approved