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Matching statistic: St000704
Mapping
St000704: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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 [[SemistandardTableaux|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.