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
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
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
Comments