Identifier
Values
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 2
[1,1,1] => 1
[4] => 1
[3,1] => 3
[2,2] => 1
[2,1,1] => 3
[1,1,1,1] => 1
[5] => 1
[4,1] => 4
[3,2] => 2
[3,1,1] => 6
[2,2,1] => 3
[2,1,1,1] => 4
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 5
[4,2] => 3
[4,1,1] => 10
[3,3] => 1
[3,2,1] => 8
[3,1,1,1] => 10
[2,2,2] => 1
[2,2,1,1] => 6
[2,1,1,1,1] => 5
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 6
[5,2] => 4
[5,1,1] => 15
[4,3] => 2
[4,2,1] => 15
[4,1,1,1] => 20
[3,3,1] => 6
[3,2,2] => 3
[3,2,1,1] => 20
[3,1,1,1,1] => 15
[2,2,2,1] => 4
[2,2,1,1,1] => 10
[2,1,1,1,1,1] => 6
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 7
[6,2] => 5
[6,1,1] => 21
[5,3] => 3
[5,2,1] => 24
[5,1,1,1] => 35
[4,4] => 1
[4,3,1] => 15
[4,2,2] => 6
[4,2,1,1] => 45
[4,1,1,1,1] => 35
[3,3,2] => 3
[3,3,1,1] => 20
[3,2,2,1] => 15
[3,2,1,1,1] => 40
[3,1,1,1,1,1] => 21
[2,2,2,2] => 1
[2,2,2,1,1] => 10
[2,2,1,1,1,1] => 15
[2,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 8
[7,2] => 6
[7,1,1] => 28
[6,3] => 4
[6,2,1] => 35
[6,1,1,1] => 56
[5,4] => 2
[5,3,1] => 27
[5,2,2] => 10
[5,2,1,1] => 84
[5,1,1,1,1] => 70
[4,4,1] => 10
[4,3,2] => 8
[4,3,1,1] => 60
[4,2,2,1] => 36
[4,2,1,1,1] => 105
[4,1,1,1,1,1] => 56
[3,3,3] => 1
[3,3,2,1] => 20
[3,3,1,1,1] => 50
[3,2,2,2] => 4
[3,2,2,1,1] => 45
[3,2,1,1,1,1] => 70
[3,1,1,1,1,1,1] => 28
[2,2,2,2,1] => 5
[2,2,2,1,1,1] => 20
[2,2,1,1,1,1,1] => 21
[2,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 9
[8,2] => 7
[8,1,1] => 36
[7,3] => 5
[7,2,1] => 48
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Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
References
[1] Fulton, W., Harris, J. Representation theory MathSciNet:1153249
Code
def statistic(L):
return SemistandardTableaux(shape=L,max_entry=len(L)).cardinality()
def statistic_alt1(L):
return prod( QQ(len(L)+j-i)/L.hook_length(i,j) for i,j in L.cells() )
def statistic_alt2(L):
return SymmetricFunctions(QQ).schur()(L).expand(len(L))([1]*len(L))
Created
Mar 07, 2017 at 09:15 by Christian Stump
Updated
Mar 07, 2017 at 09:15 by Christian Stump
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