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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-François 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: A337107 A362588 A367073 * A362787 A341200 A300130 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 14 04:13 EDT 2024. Contains 375146 sequences. (Running on oeis4.)