login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: A220905 A226573 A260693 * A300130 A101371 A325754

Adjacent sequences:  A176126 A176127 A176128 * A176130 A176131 A176132

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 29 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 6 10:14 EDT 2020. Contains 336245 sequences. (Running on oeis4.)