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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.

%I A176129 #49 Sep 21 2018 22:17:49
%S A176129 1,1,0,1,2,0,1,6,16,0,1,12,174,192,0,1,20,690,7020,2816,0,1,30,1876,
%T A176129 52808,325590,46592,0,1,42,4140,229680,4558410,16290708,835584,0,1,56,
%U A176129 7986,738192,31497284,420421056,854630476,15876096,0
%N 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.
%C A176129 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
%H A176129 Alois P. Heinz, <a href="/A176129/b176129.txt">Antidiagonals n = 0..140, flattened</a>
%H A176129 S. B. Ekhad, D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229v1 [math.CO], 2012
%H A176129 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A176129 Square array A(n,k) begins:
%e A176129   1,      1,        1,         1,          1,           1, ...
%e A176129   0,      2,        6,        12,         20,          30, ...
%e A176129   0,     16,      174,       690,       1876,        4140, ...
%e A176129   0,    192,     7020,     52808,     229680,      738192, ...
%e A176129   0,   2816,   325590,   4558410,   31497284,   146955276, ...
%e A176129   0,  46592, 16290708, 420421056, 4600393936, 31113230148, ...
%p A176129 b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y), `if`(z>x, 0,
%p A176129       `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
%p A176129       `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0))))
%p A176129     end:
%p A176129 A:= (n, k)-> b(n*k, n, n):
%p A176129 seq(seq(A(n, d-n), n=0..d), d=0..8);
%t A176129 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 *)
%Y A176129 Columns k=0-10 give: A000007, A006335, A214801, A215686, A246619, A246620, A246621, A246632, A246633, A246634, A246635.
%Y A176129 Rows n=0-3 give: A000012, A002378, A215687, A215688.
%Y A176129 Main diagonal gives: A215123.
%K A176129 nonn,tabl
%O A176129 0,5
%A A176129 _Alois P. Heinz_, Jul 29 2012