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

0,4

Comments

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

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

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

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

Row sums give: A215002.

Central row elements give: A214801.

Sequence in context: A064850 A151853 A268766 * A196201 A342982 A128045

Adjacent sequences: A214772 A214773 A214774 * A214776 A214777 A214778

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 28 2012

Status

approved