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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A226573 A260693 A337107 * A341200 A300130 A101371

Adjacent sequences:  A176126 A176127 A176128 * A176130 A176131 A176132

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 29 2012

STATUS

approved

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Last modified August 13 08:18 EDT 2022. Contains 356079 sequences. (Running on oeis4.)