Identifier
Values
[2] => [1,1] => 1
[1,1] => [2] => 1
[3] => [1,1,1] => 1
[2,1] => [2,1] => 2
[1,1,1] => [3] => 1
[4] => [1,1,1,1] => 1
[3,1] => [2,1,1] => 3
[2,2] => [2,2] => 1
[2,1,1] => [3,1] => 3
[1,1,1,1] => [4] => 1
[5] => [1,1,1,1,1] => 1
[4,1] => [2,1,1,1] => 4
[3,2] => [2,2,1] => 3
[3,1,1] => [3,1,1] => 6
[2,2,1] => [3,2] => 2
[2,1,1,1] => [4,1] => 4
[1,1,1,1,1] => [5] => 1
[6] => [1,1,1,1,1,1] => 1
[5,1] => [2,1,1,1,1] => 5
[4,2] => [2,2,1,1] => 6
[4,1,1] => [3,1,1,1] => 10
[3,3] => [2,2,2] => 1
[3,2,1] => [3,2,1] => 8
[3,1,1,1] => [4,1,1] => 10
[2,2,2] => [3,3] => 1
[2,2,1,1] => [4,2] => 3
[2,1,1,1,1] => [5,1] => 5
[1,1,1,1,1,1] => [6] => 1
[7] => [1,1,1,1,1,1,1] => 1
[6,1] => [2,1,1,1,1,1] => 6
[5,2] => [2,2,1,1,1] => 10
[5,1,1] => [3,1,1,1,1] => 15
[4,3] => [2,2,2,1] => 4
[4,2,1] => [3,2,1,1] => 20
[4,1,1,1] => [4,1,1,1] => 20
[3,3,1] => [3,2,2] => 3
[3,2,2] => [3,3,1] => 6
[3,2,1,1] => [4,2,1] => 15
[3,1,1,1,1] => [5,1,1] => 15
[2,2,2,1] => [4,3] => 2
[2,2,1,1,1] => [5,2] => 4
[2,1,1,1,1,1] => [6,1] => 6
[1,1,1,1,1,1,1] => [7] => 1
[8] => [1,1,1,1,1,1,1,1] => 1
[7,1] => [2,1,1,1,1,1,1] => 7
[6,2] => [2,2,1,1,1,1] => 15
[6,1,1] => [3,1,1,1,1,1] => 21
[5,3] => [2,2,2,1,1] => 10
[5,2,1] => [3,2,1,1,1] => 40
[5,1,1,1] => [4,1,1,1,1] => 35
[4,4] => [2,2,2,2] => 1
[4,3,1] => [3,2,2,1] => 15
[4,2,2] => [3,3,1,1] => 20
[4,2,1,1] => [4,2,1,1] => 45
[4,1,1,1,1] => [5,1,1,1] => 35
[3,3,2] => [3,3,2] => 3
[3,3,1,1] => [4,2,2] => 6
[3,2,2,1] => [4,3,1] => 15
[3,2,1,1,1] => [5,2,1] => 24
[3,1,1,1,1,1] => [6,1,1] => 21
[2,2,2,2] => [4,4] => 1
[2,2,2,1,1] => [5,3] => 3
[2,2,1,1,1,1] => [6,2] => 5
[2,1,1,1,1,1,1] => [7,1] => 7
[1,1,1,1,1,1,1,1] => [8] => 1
[9] => [1,1,1,1,1,1,1,1,1] => 1
[8,1] => [2,1,1,1,1,1,1,1] => 8
[7,2] => [2,2,1,1,1,1,1] => 21
[7,1,1] => [3,1,1,1,1,1,1] => 28
[6,3] => [2,2,2,1,1,1] => 20
[6,2,1] => [3,2,1,1,1,1] => 70
[6,1,1,1] => [4,1,1,1,1,1] => 56
[5,4] => [2,2,2,2,1] => 5
[5,3,1] => [3,2,2,1,1] => 45
[5,2,2] => [3,3,1,1,1] => 50
[5,2,1,1] => [4,2,1,1,1] => 105
[5,1,1,1,1] => [5,1,1,1,1] => 70
[4,4,1] => [3,2,2,2] => 4
[4,3,2] => [3,3,2,1] => 20
[4,3,1,1] => [4,2,2,1] => 36
[4,2,2,1] => [4,3,1,1] => 60
[4,2,1,1,1] => [5,2,1,1] => 84
[4,1,1,1,1,1] => [6,1,1,1] => 56
[3,3,3] => [3,3,3] => 1
[3,3,2,1] => [4,3,2] => 8
[3,3,1,1,1] => [5,2,2] => 10
[3,2,2,2] => [4,4,1] => 10
[3,2,2,1,1] => [5,3,1] => 27
[3,2,1,1,1,1] => [6,2,1] => 35
[3,1,1,1,1,1,1] => [7,1,1] => 28
[2,2,2,2,1] => [5,4] => 2
[2,2,2,1,1,1] => [6,3] => 4
[2,2,1,1,1,1,1] => [7,2] => 6
[2,1,1,1,1,1,1,1] => [8,1] => 8
[1,1,1,1,1,1,1,1,1] => [9] => 1
[10] => [1,1,1,1,1,1,1,1,1,1] => 1
[9,1] => [2,1,1,1,1,1,1,1,1] => 9
[8,2] => [2,2,1,1,1,1,1,1] => 28
[8,1,1] => [3,1,1,1,1,1,1,1] => 36
[7,3] => [2,2,2,1,1,1,1] => 35
[7,2,1] => [3,2,1,1,1,1,1] => 112
>>> Load all 270 entries. <<<
[7,1,1,1] => [4,1,1,1,1,1,1] => 84
[6,4] => [2,2,2,2,1,1] => 15
[6,3,1] => [3,2,2,1,1,1] => 105
[6,2,2] => [3,3,1,1,1,1] => 105
[6,2,1,1] => [4,2,1,1,1,1] => 210
[6,1,1,1,1] => [5,1,1,1,1,1] => 126
[5,5] => [2,2,2,2,2] => 1
[5,4,1] => [3,2,2,2,1] => 24
[5,3,2] => [3,3,2,1,1] => 75
[5,3,1,1] => [4,2,2,1,1] => 126
[5,2,2,1] => [4,3,1,1,1] => 175
[5,2,1,1,1] => [5,2,1,1,1] => 224
[5,1,1,1,1,1] => [6,1,1,1,1] => 126
[4,4,2] => [3,3,2,2] => 6
[4,4,1,1] => [4,2,2,2] => 10
[4,3,3] => [3,3,3,1] => 10
[4,3,2,1] => [4,3,2,1] => 64
[4,3,1,1,1] => [5,2,2,1] => 70
[4,2,2,2] => [4,4,1,1] => 50
[4,2,2,1,1] => [5,3,1,1] => 126
[4,2,1,1,1,1] => [6,2,1,1] => 140
[4,1,1,1,1,1,1] => [7,1,1,1] => 84
[3,3,3,1] => [4,3,3] => 3
[3,3,2,2] => [4,4,2] => 6
[3,3,2,1,1] => [5,3,2] => 15
[3,3,1,1,1,1] => [6,2,2] => 15
[3,2,2,2,1] => [5,4,1] => 24
[3,2,2,1,1,1] => [6,3,1] => 42
[3,2,1,1,1,1,1] => [7,2,1] => 48
[3,1,1,1,1,1,1,1] => [8,1,1] => 36
[2,2,2,2,2] => [5,5] => 1
[2,2,2,2,1,1] => [6,4] => 3
[2,2,2,1,1,1,1] => [7,3] => 5
[2,2,1,1,1,1,1,1] => [8,2] => 7
[2,1,1,1,1,1,1,1,1] => [9,1] => 9
[1,1,1,1,1,1,1,1,1,1] => [10] => 1
[11] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[10,1] => [2,1,1,1,1,1,1,1,1,1] => 10
[9,2] => [2,2,1,1,1,1,1,1,1] => 36
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 45
[8,3] => [2,2,2,1,1,1,1,1] => 56
[8,2,1] => [3,2,1,1,1,1,1,1] => 168
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 120
[7,4] => [2,2,2,2,1,1,1] => 35
[7,3,1] => [3,2,2,1,1,1,1] => 210
[7,2,2] => [3,3,1,1,1,1,1] => 196
[7,2,1,1] => [4,2,1,1,1,1,1] => 378
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 210
[6,5] => [2,2,2,2,2,1] => 6
[6,4,1] => [3,2,2,2,1,1] => 84
[6,3,2] => [3,3,2,1,1,1] => 210
[6,3,1,1] => [4,2,2,1,1,1] => 336
[6,2,2,1] => [4,3,1,1,1,1] => 420
[6,2,1,1,1] => [5,2,1,1,1,1] => 504
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 252
[5,5,1] => [3,2,2,2,2] => 5
[5,4,2] => [3,3,2,2,1] => 45
[5,4,1,1] => [4,2,2,2,1] => 70
[5,3,3] => [3,3,3,1,1] => 50
[5,3,2,1] => [4,3,2,1,1] => 280
[5,3,1,1,1] => [5,2,2,1,1] => 280
[5,2,2,2] => [4,4,1,1,1] => 175
[5,2,2,1,1] => [5,3,1,1,1] => 420
[5,2,1,1,1,1] => [6,2,1,1,1] => 420
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 210
[4,4,3] => [3,3,3,2] => 4
[4,4,2,1] => [4,3,2,2] => 20
[4,4,1,1,1] => [5,2,2,2] => 20
[4,3,3,1] => [4,3,3,1] => 36
[4,3,2,2] => [4,4,2,1] => 60
[4,3,2,1,1] => [5,3,2,1] => 140
[4,3,1,1,1,1] => [6,2,2,1] => 120
[4,2,2,2,1] => [5,4,1,1] => 140
[4,2,2,1,1,1] => [6,3,1,1] => 224
[4,2,1,1,1,1,1] => [7,2,1,1] => 216
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 120
[3,3,3,2] => [4,4,3] => 3
[3,3,3,1,1] => [5,3,3] => 6
[3,3,2,2,1] => [5,4,2] => 15
[3,3,2,1,1,1] => [6,3,2] => 24
[3,3,1,1,1,1,1] => [7,2,2] => 21
[3,2,2,2,2] => [5,5,1] => 15
[3,2,2,2,1,1] => [6,4,1] => 42
[3,2,2,1,1,1,1] => [7,3,1] => 60
[3,2,1,1,1,1,1,1] => [8,2,1] => 63
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 45
[2,2,2,2,2,1] => [6,5] => 2
[2,2,2,2,1,1,1] => [7,4] => 4
[2,2,2,1,1,1,1,1] => [8,3] => 6
[2,2,1,1,1,1,1,1,1] => [9,2] => 8
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 10
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 1
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 11
[10,2] => [2,2,1,1,1,1,1,1,1,1] => 45
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 55
[9,3] => [2,2,2,1,1,1,1,1,1] => 84
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 240
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 165
[8,4] => [2,2,2,2,1,1,1,1] => 70
[8,3,1] => [3,2,2,1,1,1,1,1] => 378
[8,2,2] => [3,3,1,1,1,1,1,1] => 336
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 630
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 330
[7,5] => [2,2,2,2,2,1,1] => 21
[7,4,1] => [3,2,2,2,1,1,1] => 224
[7,3,2] => [3,3,2,1,1,1,1] => 490
[7,3,1,1] => [4,2,2,1,1,1,1] => 756
[7,2,2,1] => [4,3,1,1,1,1,1] => 882
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 1008
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 462
[6,6] => [2,2,2,2,2,2] => 1
[6,5,1] => [3,2,2,2,2,1] => 35
[6,4,2] => [3,3,2,2,1,1] => 189
[6,4,1,1] => [4,2,2,2,1,1] => 280
[6,3,3] => [3,3,3,1,1,1] => 175
[6,3,2,1] => [4,3,2,1,1,1] => 896
[6,3,1,1,1] => [5,2,2,1,1,1] => 840
[6,2,2,2] => [4,4,1,1,1,1] => 490
[6,2,2,1,1] => [5,3,1,1,1,1] => 1134
[6,2,1,1,1,1] => [6,2,1,1,1,1] => 1050
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => 462
[5,5,2] => [3,3,2,2,2] => 10
[5,5,1,1] => [4,2,2,2,2] => 15
[5,4,3] => [3,3,3,2,1] => 40
[5,4,2,1] => [4,3,2,2,1] => 175
[5,4,1,1,1] => [5,2,2,2,1] => 160
[5,3,3,1] => [4,3,3,1,1] => 210
[5,3,2,2] => [4,4,2,1,1] => 315
[5,3,2,1,1] => [5,3,2,1,1] => 700
[5,3,1,1,1,1] => [6,2,2,1,1] => 540
[5,2,2,2,1] => [5,4,1,1,1] => 560
[5,2,2,1,1,1] => [6,3,1,1,1] => 840
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 720
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 330
[4,4,4] => [3,3,3,3] => 1
[4,4,3,1] => [4,3,3,2] => 15
[4,4,2,2] => [4,4,2,2] => 20
[4,4,2,1,1] => [5,3,2,2] => 45
[4,4,1,1,1,1] => [6,2,2,2] => 35
[4,3,3,2] => [4,4,3,1] => 45
[4,3,3,1,1] => [5,3,3,1] => 84
[4,3,2,2,1] => [5,4,2,1] => 175
[4,3,2,1,1,1] => [6,3,2,1] => 256
[4,3,1,1,1,1,1] => [7,2,2,1] => 189
[4,2,2,2,2] => [5,5,1,1] => 105
[4,2,2,2,1,1] => [6,4,1,1] => 280
[4,2,2,1,1,1,1] => [7,3,1,1] => 360
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 315
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 165
[3,3,3,3] => [4,4,4] => 1
[3,3,3,2,1] => [5,4,3] => 8
[3,3,3,1,1,1] => [6,3,3] => 10
[3,3,2,2,2] => [5,5,2] => 10
[3,3,2,2,1,1] => [6,4,2] => 27
[3,3,2,1,1,1,1] => [7,3,2] => 35
[3,3,1,1,1,1,1,1] => [8,2,2] => 28
[3,2,2,2,2,1] => [6,5,1] => 35
[3,2,2,2,1,1,1] => [7,4,1] => 64
[3,2,2,1,1,1,1,1] => [8,3,1] => 81
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 80
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 55
[2,2,2,2,2,2] => [6,6] => 1
[2,2,2,2,2,1,1] => [7,5] => 3
[2,2,2,2,1,1,1,1] => [8,4] => 5
[2,2,2,1,1,1,1,1,1] => [9,3] => 7
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 9
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 11
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 1
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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.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.