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A215122 Number T(n,k) of solid standard Young tableaux of shape [[(n-k)*k,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A215122 #28 Sep 21 2018 22:18:21
%S A215122 1,0,0,0,2,0,0,8,8,0,0,30,174,30,0,0,112,2084,2084,112,0,0,420,21025,
%T A215122 52808,21025,420,0,0,1584,194064,994788,994788,194064,1584,0,0,6006,
%U A215122 1694224,16074586,31497284,16074586,1694224,6006,0
%N A215122 Number T(n,k) of solid standard Young tableaux of shape [[(n-k)*k,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A215122 Alois P. Heinz, <a href="/A215122/b215122.txt">Rows n = 0..140, flattened</a>
%H A215122 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 A215122 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A215122 Triangle T(n,k) begins:
%e A215122   1;
%e A215122   0,    0;
%e A215122   0,    2,      0;
%e A215122   0,    8,      8,      0;
%e A215122   0,   30,    174,     30,      0;
%e A215122   0,  112,   2084,   2084,    112,      0;
%e A215122   0,  420,  21025,  52808,  21025,    420,    0;
%e A215122   0, 1584, 194064, 994788, 994788, 194064, 1584,  0;
%p A215122 b:= proc(x, y, z) option remember; `if`(z<y, b(x, z, y),
%p A215122       `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
%p A215122       `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
%p A215122     end:
%p A215122 T:= (n, k)-> `if`(k=0 xor k=n, 0, b((n-k)*k, k, n-k)):
%p A215122 seq(seq(T(n, k), k=0..n), n=0..10);
%t A215122 b[x_, y_, z_] := b[x, y, z] = If[z<y, b[x, z, y], 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]]]; T[n_, k_] := If[k == 0 || k == n, 0, b[(n-k)*k, k, n-k]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 19 2015, after _Alois P. Heinz_ *)
%Y A215122 Columns k=0-2 give: A000007, A162551(n-1), A215124.
%Y A215122 Central elements of rows give A215123.
%K A215122 nonn,tabl
%O A215122 0,5
%A A215122 _Alois P. Heinz_, Aug 03 2012