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
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
Programs
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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 *)
Comments