A214775 Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 2, 6, 2, 5, 25, 25, 5, 14, 98, 174, 98, 14, 42, 378, 962, 962, 378, 42, 132, 1452, 4804, 7020, 4804, 1452, 132, 429, 5577, 22689, 43573, 43573, 22689, 5577, 429, 1430, 21450, 103510, 245962, 325590, 245962, 103510, 21450, 1430

Offset: 0

Author

Alois P. Heinz, Jul 28 2012

Keywords

• nonn
• tabl

Comments

T(n,k) is odd if and only if n = 2^i-1 for i in {0, 1, 2, ... } = A001477.

Examples

			Triangle T(n,k) begins:
    1;
    1,    1;
    2,    6,    2;
    5,   25,   25,    5;
   14,   98,  174,   98,   14;
   42,  378,  962,  962,  378,   42;
  132, 1452, 4804, 7020, 4804, 1452, 132;
  ...
		

Links

• Alois P. Heinz, Rows 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

Columns 0-5 give: A000108, A214955, A215298, A215299, A215300, A215301.

Row sums give: A215002.

Central row elements give: A214801.

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:
T:= (n, k)-> b(n, k, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);

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]; Table[T[n, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

Sage

@CachedFunction
def B(x, y, z) :
    if z > y : return B(x, z, y)
    if x==y and y==z and z==0 : return 1
    a = B(x-1, y, z) if x>y and x>z else 0
    b = B(x, y-1, z) if y>0 else 0
    c = B(x, y, z-1) if z>0 else 0
    return a + b + c
T = lambda n, k: B(n, k, n-k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
# After Maple code of Alois P. Heinz. Peter Luschny, Jul 30 2012