A215002 Number of all solid standard Young tableaux of shape [[n,k],[n-k]] for 0<=k<=n.

1, 2, 10, 60, 398, 2764, 19796, 144536, 1070294, 8007052, 60380940, 458185992, 3494554380, 26764583096, 205711091880, 1585822364592, 12256625999718, 94942581080204, 736895626109148, 5729374337686696, 44615143884080996, 347905737091032552, 2716349710039969688

Offset: 0

Author

Alois P. Heinz, Jul 30 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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

Row sums of A214775.

Programs

Maple

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
      `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-> add(b(n, k, n-k), k=0..n):
seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 2, 10, 60][n+1],
      ((1640*n^8 -1180*n^7 -7114*n^6 +5615*n^5 +20240*n^4 -35170*n^3
       +20379*n^2 -4050*n) *a(n-1) +(-7640*n^8 +14560*n^7 +47374*n^6
       -140900*n^5 -37160*n^4 +601810*n^3 -944154*n^2 +580680*n -113400)
       *a(n-2) +(-28800*n^8 +181440*n^7 -138240*n^6 -874800*n^5 +670680*n^4
       +3165480*n^3 -3646440*n^2 -12960*n -453600) *a(n-3) +(207360*n^8
       -1451520*n^7 +912384*n^6 [phone]*n^5 -15720480*n^4 -42042240*n^3
       [phone]*n^2 -50388480*n [phone]) *a(n-4)) / (n* (n+1) *(2*n-1)
       *(2*n+1) *(20*n^4-47*n^2-33*n+90)))
    end:
seq(a(n), n=0..25);

Mathematica

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_] := b[n, k, n-k]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Formula

See Maple programs.

a(n) ~ sqrt((5*sqrt(5))/2-11/2) * 8^n / sqrt(Pi*n). - Vaclav Kotesovec, Jul 16 2014