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Search: a214801 -id:a214801
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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.
+10
14
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
In general, column k is (for k > 1) asymptotic to sqrt((k+2)*(k^2 - 20*k - 8 + sqrt(k*(k+8)^3)) / (8*k^3)) * ((k+2)^(k+2)/k^k)^n / (Pi*n). - Vaclav Kotesovec, Aug 31 2014
LINKS
Alois P. Heinz, Antidiagonals 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
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, ...
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
Columns k=0-10 give: A000007, A006335, A214801, A215686, A246619, A246620, A246621, A246632, A246633, A246634, A246635.
Rows n=0-3 give: A000012, A002378, A215687, A215688.
Main diagonal gives: A215123.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 29 2012
STATUS
approved
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.
+10
8
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
(list; table; graph; refs; listen; history; text; internal format)
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.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 28 2012
STATUS
approved
A246619
Number of solid standard Young tableaux of shape [[4*n,n],[n]].
+10
4
1, 20, 1876, 229680, 31497284, 4600393936, 699440711760, 109341854545792, 17445620031680100, 2827280025640259280, 463882742476664594512, 76875122571167921990080, 12845419277094419018993808, 2161338658294952555703260480, 365816910931667192749720139072
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
a(n) ~ sqrt(4*sqrt(3)-6) * 3^(6*n+1) / (Pi * n * 2^(2*n+3)). - Vaclav Kotesovec, Aug 31 2014
CROSSREFS
Cf. column k=4 of A176129.
Cf. A214801, A215686.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 31 2014
STATUS
approved
A246620
Number of solid standard Young tableaux of shape [[5*n,n],[n]].
+10
3
1, 30, 4140, 738192, 146955276, 31113230148, 6851807953900, 1550766110966400, 358116337203378732, 83984165552626389864, 19937272615715693766528, 4779986445560522545646400, 1155414579663560935856564700, 281212253617692376239817669056
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..150
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
a(n) ~ sqrt(7/10*(13*sqrt(65)-83))/10 * 7^(7*n) / (Pi * n * 5^(5*n)). - Vaclav Kotesovec, Aug 31 2014
CROSSREFS
Cf. Column k=5 of A176129.
Cf. A214801, A215686, A246619.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 31 2014
STATUS
approved
A246621
Number of solid standard Young tableaux of shape [[6*n,n],[n]].
+10
2
1, 42, 7986, 1950512, 530931786, 153580152492, 46190668836656, 14274134610246720, 4500027052542851130, 1440557297650459814996, 466776334221187994469180, 152741149363060061495819904, 50388989722150284436348268528, 16737346518387797143628281698720
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..150
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
a(n) ~ sqrt((7*sqrt(21)-23)/6)/3 * 8^(8*n) / (Pi * n * 6^(6*n)). - Vaclav Kotesovec, Aug 31 2014
CROSSREFS
Cf. column k=6 of A176129.
Cf. A214801, A215686, A246619, A246620.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 31 2014
STATUS
approved
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