Identifier
Values
[1] => [1] => 0
[2] => [1,1] => 1
[1,1] => [2] => 0
[3] => [1,1,1] => 3
[2,1] => [2,1] => 1
[1,1,1] => [3] => 0
[4] => [1,1,1,1] => 6
[3,1] => [2,1,1] => 3
[2,2] => [2,2] => 2
[2,1,1] => [3,1] => 1
[1,1,1,1] => [4] => 0
[5] => [1,1,1,1,1] => 10
[4,1] => [2,1,1,1] => 6
[3,2] => [2,2,1] => 4
[3,1,1] => [3,1,1] => 3
[2,2,1] => [3,2] => 2
[2,1,1,1] => [4,1] => 1
[1,1,1,1,1] => [5] => 0
[6] => [1,1,1,1,1,1] => 15
[5,1] => [2,1,1,1,1] => 10
[4,2] => [2,2,1,1] => 7
[4,1,1] => [3,1,1,1] => 6
[3,3] => [2,2,2] => 6
[3,2,1] => [3,2,1] => 4
[3,1,1,1] => [4,1,1] => 3
[2,2,2] => [3,3] => 3
[2,2,1,1] => [4,2] => 2
[2,1,1,1,1] => [5,1] => 1
[1,1,1,1,1,1] => [6] => 0
[7] => [1,1,1,1,1,1,1] => 21
[6,1] => [2,1,1,1,1,1] => 15
[5,2] => [2,2,1,1,1] => 11
[5,1,1] => [3,1,1,1,1] => 10
[4,3] => [2,2,2,1] => 9
[4,2,1] => [3,2,1,1] => 7
[4,1,1,1] => [4,1,1,1] => 6
[3,3,1] => [3,2,2] => 6
[3,2,2] => [3,3,1] => 5
[3,2,1,1] => [4,2,1] => 4
[3,1,1,1,1] => [5,1,1] => 3
[2,2,2,1] => [4,3] => 3
[2,2,1,1,1] => [5,2] => 2
[2,1,1,1,1,1] => [6,1] => 1
[1,1,1,1,1,1,1] => [7] => 0
[8] => [1,1,1,1,1,1,1,1] => 28
[7,1] => [2,1,1,1,1,1,1] => 21
[6,2] => [2,2,1,1,1,1] => 16
[6,1,1] => [3,1,1,1,1,1] => 15
[5,3] => [2,2,2,1,1] => 13
[5,2,1] => [3,2,1,1,1] => 11
[5,1,1,1] => [4,1,1,1,1] => 10
[4,4] => [2,2,2,2] => 12
[4,3,1] => [3,2,2,1] => 9
[4,2,2] => [3,3,1,1] => 8
[4,2,1,1] => [4,2,1,1] => 7
[4,1,1,1,1] => [5,1,1,1] => 6
[3,3,2] => [3,3,2] => 7
[3,3,1,1] => [4,2,2] => 6
[3,2,2,1] => [4,3,1] => 5
[3,2,1,1,1] => [5,2,1] => 4
[3,1,1,1,1,1] => [6,1,1] => 3
[2,2,2,2] => [4,4] => 4
[2,2,2,1,1] => [5,3] => 3
[2,2,1,1,1,1] => [6,2] => 2
[2,1,1,1,1,1,1] => [7,1] => 1
[1,1,1,1,1,1,1,1] => [8] => 0
[9] => [1,1,1,1,1,1,1,1,1] => 36
[8,1] => [2,1,1,1,1,1,1,1] => 28
[7,2] => [2,2,1,1,1,1,1] => 22
[7,1,1] => [3,1,1,1,1,1,1] => 21
[6,3] => [2,2,2,1,1,1] => 18
[6,2,1] => [3,2,1,1,1,1] => 16
[6,1,1,1] => [4,1,1,1,1,1] => 15
[5,4] => [2,2,2,2,1] => 16
[5,3,1] => [3,2,2,1,1] => 13
[5,2,2] => [3,3,1,1,1] => 12
[5,2,1,1] => [4,2,1,1,1] => 11
[5,1,1,1,1] => [5,1,1,1,1] => 10
[4,4,1] => [3,2,2,2] => 12
[4,3,2] => [3,3,2,1] => 10
[4,3,1,1] => [4,2,2,1] => 9
[4,2,2,1] => [4,3,1,1] => 8
[4,2,1,1,1] => [5,2,1,1] => 7
[4,1,1,1,1,1] => [6,1,1,1] => 6
[3,3,3] => [3,3,3] => 9
[3,3,2,1] => [4,3,2] => 7
[3,3,1,1,1] => [5,2,2] => 6
[3,2,2,2] => [4,4,1] => 6
[3,2,2,1,1] => [5,3,1] => 5
[3,2,1,1,1,1] => [6,2,1] => 4
[3,1,1,1,1,1,1] => [7,1,1] => 3
[2,2,2,2,1] => [5,4] => 4
[2,2,2,1,1,1] => [6,3] => 3
[2,2,1,1,1,1,1] => [7,2] => 2
[2,1,1,1,1,1,1,1] => [8,1] => 1
[1,1,1,1,1,1,1,1,1] => [9] => 0
[10] => [1,1,1,1,1,1,1,1,1,1] => 45
[9,1] => [2,1,1,1,1,1,1,1,1] => 36
[8,2] => [2,2,1,1,1,1,1,1] => 29
[8,1,1] => [3,1,1,1,1,1,1,1] => 28
[7,3] => [2,2,2,1,1,1,1] => 24
>>> Load all 238 entries. <<<
[7,2,1] => [3,2,1,1,1,1,1] => 22
[7,1,1,1] => [4,1,1,1,1,1,1] => 21
[6,4] => [2,2,2,2,1,1] => 21
[6,3,1] => [3,2,2,1,1,1] => 18
[6,2,2] => [3,3,1,1,1,1] => 17
[6,2,1,1] => [4,2,1,1,1,1] => 16
[6,1,1,1,1] => [5,1,1,1,1,1] => 15
[5,5] => [2,2,2,2,2] => 20
[5,4,1] => [3,2,2,2,1] => 16
[5,3,2] => [3,3,2,1,1] => 14
[5,3,1,1] => [4,2,2,1,1] => 13
[5,2,2,1] => [4,3,1,1,1] => 12
[5,2,1,1,1] => [5,2,1,1,1] => 11
[5,1,1,1,1,1] => [6,1,1,1,1] => 10
[4,4,2] => [3,3,2,2] => 13
[4,4,1,1] => [4,2,2,2] => 12
[4,3,3] => [3,3,3,1] => 12
[4,3,2,1] => [4,3,2,1] => 10
[4,3,1,1,1] => [5,2,2,1] => 9
[4,2,2,2] => [4,4,1,1] => 9
[4,2,2,1,1] => [5,3,1,1] => 8
[4,2,1,1,1,1] => [6,2,1,1] => 7
[4,1,1,1,1,1,1] => [7,1,1,1] => 6
[3,3,3,1] => [4,3,3] => 9
[3,3,2,2] => [4,4,2] => 8
[3,3,2,1,1] => [5,3,2] => 7
[3,3,1,1,1,1] => [6,2,2] => 6
[3,2,2,2,1] => [5,4,1] => 6
[3,2,2,1,1,1] => [6,3,1] => 5
[3,2,1,1,1,1,1] => [7,2,1] => 4
[3,1,1,1,1,1,1,1] => [8,1,1] => 3
[2,2,2,2,2] => [5,5] => 5
[2,2,2,2,1,1] => [6,4] => 4
[2,2,2,1,1,1,1] => [7,3] => 3
[2,2,1,1,1,1,1,1] => [8,2] => 2
[2,1,1,1,1,1,1,1,1] => [9,1] => 1
[1,1,1,1,1,1,1,1,1,1] => [10] => 0
[6,5] => [2,2,2,2,2,1] => 25
[5,5,1] => [3,2,2,2,2] => 20
[5,4,2] => [3,3,2,2,1] => 17
[5,4,1,1] => [4,2,2,2,1] => 16
[5,3,3] => [3,3,3,1,1] => 16
[5,3,2,1] => [4,3,2,1,1] => 14
[5,3,1,1,1] => [5,2,2,1,1] => 13
[5,2,2,2] => [4,4,1,1,1] => 13
[5,2,2,1,1] => [5,3,1,1,1] => 12
[4,4,3] => [3,3,3,2] => 15
[4,4,2,1] => [4,3,2,2] => 13
[4,4,1,1,1] => [5,2,2,2] => 12
[4,3,3,1] => [4,3,3,1] => 12
[4,3,2,2] => [4,4,2,1] => 11
[4,3,2,1,1] => [5,3,2,1] => 10
[4,2,2,2,1] => [5,4,1,1] => 9
[3,3,3,2] => [4,4,3] => 10
[3,3,3,1,1] => [5,3,3] => 9
[3,3,2,2,1] => [5,4,2] => 8
[3,2,2,2,2] => [5,5,1] => 7
[3,2,2,2,1,1] => [6,4,1] => 6
[2,2,2,2,2,1] => [6,5] => 5
[2,2,2,2,1,1,1] => [7,4] => 4
[2,2,2,1,1,1,1,1] => [8,3] => 3
[6,6] => [2,2,2,2,2,2] => 30
[6,4,2] => [3,3,2,2,1,1] => 22
[5,5,2] => [3,3,2,2,2] => 21
[5,4,3] => [3,3,3,2,1] => 19
[5,4,2,1] => [4,3,2,2,1] => 17
[5,4,1,1,1] => [5,2,2,2,1] => 16
[5,3,3,1] => [4,3,3,1,1] => 16
[5,3,2,2] => [4,4,2,1,1] => 15
[5,3,2,1,1] => [5,3,2,1,1] => 14
[5,2,2,2,1] => [5,4,1,1,1] => 13
[4,4,4] => [3,3,3,3] => 18
[4,4,3,1] => [4,3,3,2] => 15
[4,4,2,2] => [4,4,2,2] => 14
[4,4,2,1,1] => [5,3,2,2] => 13
[4,3,3,2] => [4,4,3,1] => 13
[4,3,3,1,1] => [5,3,3,1] => 12
[4,3,2,2,1] => [5,4,2,1] => 11
[3,3,3,3] => [4,4,4] => 12
[3,3,3,2,1] => [5,4,3] => 10
[3,3,2,2,2] => [5,5,2] => 9
[3,3,2,2,1,1] => [6,4,2] => 8
[3,2,2,2,1,1,1] => [7,4,1] => 6
[2,2,2,2,2,2] => [6,6] => 6
[2,2,2,2,2,1,1] => [7,5] => 5
[5,5,3] => [3,3,3,2,2] => 23
[5,4,4] => [3,3,3,3,1] => 22
[5,4,3,1] => [4,3,3,2,1] => 19
[5,4,2,2] => [4,4,2,2,1] => 18
[5,4,2,1,1] => [5,3,2,2,1] => 17
[5,3,3,2] => [4,4,3,1,1] => 17
[5,3,3,1,1] => [5,3,3,1,1] => 16
[5,3,2,2,1] => [5,4,2,1,1] => 15
[4,4,4,1] => [4,3,3,3] => 18
[4,4,3,2] => [4,4,3,2] => 16
[4,4,3,1,1] => [5,3,3,2] => 15
[4,4,2,2,1] => [5,4,2,2] => 14
[4,3,3,3] => [4,4,4,1] => 15
[4,3,3,2,1] => [5,4,3,1] => 13
[3,3,3,3,1] => [5,4,4] => 12
[3,3,3,2,2] => [5,5,3] => 11
[3,3,2,2,1,1,1] => [7,4,2] => 8
[3,2,2,2,2,1,1] => [7,5,1] => 7
[2,2,2,2,2,1,1,1] => [8,5] => 5
[5,5,4] => [3,3,3,3,2] => 26
[5,5,2,1,1] => [5,3,2,2,2] => 21
[5,4,3,2] => [4,4,3,2,1] => 20
[5,4,3,1,1] => [5,3,3,2,1] => 19
[5,4,2,2,1] => [5,4,2,2,1] => 18
[5,3,3,2,1] => [5,4,3,1,1] => 17
[4,4,4,2] => [4,4,3,3] => 19
[4,4,3,3] => [4,4,4,2] => 18
[4,4,3,2,1] => [5,4,3,2] => 16
[3,3,3,3,2] => [5,5,4] => 13
[3,3,3,2,1,1,1] => [7,4,3] => 10
[3,3,2,2,2,1,1] => [7,5,2] => 9
[3,2,2,2,2,1,1,1] => [8,5,1] => 7
[2,2,2,2,2,1,1,1,1] => [9,5] => 5
[6,5,2,1,1] => [5,3,2,2,2,1] => 26
[5,5,5] => [3,3,3,3,3] => 30
[5,4,3,2,1] => [5,4,3,2,1] => 20
[4,4,4,3] => [4,4,4,3] => 21
[3,3,3,3,3] => [5,5,5] => 15
[3,3,3,2,2,1,1] => [7,5,3] => 11
[3,3,2,2,2,1,1,1] => [8,5,2] => 9
[3,2,2,2,2,1,1,1,1] => [9,5,1] => 7
[4,4,4,4] => [4,4,4,4] => 24
[4,3,3,2,2,1,1] => [7,5,3,1] => 14
[3,3,3,2,2,1,1,1] => [8,5,3] => 11
[3,3,3,2,2,2,1,1] => [8,6,3] => 12
[] => [] => 0
[6,5,4,3,2,1] => [6,5,4,3,2,1] => 35
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 56
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 84
[3,3,3,3,2,2,1,1,1,1] => [10,6,4] => 14
[4,4,3,3,2,2,1,1] => [8,6,4,2] => 20
[5,5,4,4,3,3,2,2,1,1] => [10,8,6,4,2] => 40
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Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
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.