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.
Original entry on oeis.org
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
Offset: 0
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, ...
Columns k=0-10 give:
A000007,
A006335,
A214801,
A215686,
A246619,
A246620,
A246621,
A246632,
A246633,
A246634,
A246635.
-
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);
-
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 *)
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.
Original entry on oeis.org
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
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;
...
Central row elements give:
A214801.
-
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);
-
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 *)
-
@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
A246619
Number of solid standard Young tableaux of shape [[4*n,n],[n]].
Original entry on oeis.org
1, 20, 1876, 229680, 31497284, 4600393936, 699440711760, 109341854545792, 17445620031680100, 2827280025640259280, 463882742476664594512, 76875122571167921990080, 12845419277094419018993808, 2161338658294952555703260480, 365816910931667192749720139072
Offset: 0
A246620
Number of solid standard Young tableaux of shape [[5*n,n],[n]].
Original entry on oeis.org
1, 30, 4140, 738192, 146955276, 31113230148, 6851807953900, 1550766110966400, 358116337203378732, 83984165552626389864, 19937272615715693766528, 4779986445560522545646400, 1155414579663560935856564700, 281212253617692376239817669056
Offset: 0
A246621
Number of solid standard Young tableaux of shape [[6*n,n],[n]].
Original entry on oeis.org
1, 42, 7986, 1950512, 530931786, 153580152492, 46190668836656, 14274134610246720, 4500027052542851130, 1440557297650459814996, 466776334221187994469180, 152741149363060061495819904, 50388989722150284436348268528, 16737346518387797143628281698720
Offset: 0
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