Identifier
Values
[2] => [1,1] => 0
[1,1] => [2] => 1
[3] => [1,1,1] => 0
[2,1] => [2,1] => 1
[1,1,1] => [3] => 3
[4] => [1,1,1,1] => 0
[3,1] => [2,1,1] => 1
[2,2] => [2,2] => 2
[2,1,1] => [3,1] => 3
[1,1,1,1] => [4] => 6
[5] => [1,1,1,1,1] => 0
[4,1] => [2,1,1,1] => 1
[3,2] => [2,2,1] => 2
[3,1,1] => [3,1,1] => 3
[2,2,1] => [3,2] => 4
[2,1,1,1] => [4,1] => 6
[1,1,1,1,1] => [5] => 10
[6] => [1,1,1,1,1,1] => 0
[5,1] => [2,1,1,1,1] => 1
[4,2] => [2,2,1,1] => 2
[4,1,1] => [3,1,1,1] => 3
[3,3] => [2,2,2] => 3
[3,2,1] => [3,2,1] => 4
[3,1,1,1] => [4,1,1] => 6
[2,2,2] => [3,3] => 6
[2,2,1,1] => [4,2] => 7
[2,1,1,1,1] => [5,1] => 10
[1,1,1,1,1,1] => [6] => 15
[7] => [1,1,1,1,1,1,1] => 0
[6,1] => [2,1,1,1,1,1] => 1
[5,2] => [2,2,1,1,1] => 2
[5,1,1] => [3,1,1,1,1] => 3
[4,3] => [2,2,2,1] => 3
[4,2,1] => [3,2,1,1] => 4
[4,1,1,1] => [4,1,1,1] => 6
[3,3,1] => [3,2,2] => 5
[3,2,2] => [3,3,1] => 6
[3,2,1,1] => [4,2,1] => 7
[3,1,1,1,1] => [5,1,1] => 10
[2,2,2,1] => [4,3] => 9
[2,2,1,1,1] => [5,2] => 11
[2,1,1,1,1,1] => [6,1] => 15
[1,1,1,1,1,1,1] => [7] => 21
[8] => [1,1,1,1,1,1,1,1] => 0
[7,1] => [2,1,1,1,1,1,1] => 1
[6,2] => [2,2,1,1,1,1] => 2
[6,1,1] => [3,1,1,1,1,1] => 3
[5,3] => [2,2,2,1,1] => 3
[5,2,1] => [3,2,1,1,1] => 4
[5,1,1,1] => [4,1,1,1,1] => 6
[4,4] => [2,2,2,2] => 4
[4,3,1] => [3,2,2,1] => 5
[4,2,2] => [3,3,1,1] => 6
[4,2,1,1] => [4,2,1,1] => 7
[4,1,1,1,1] => [5,1,1,1] => 10
[3,3,2] => [3,3,2] => 7
[3,3,1,1] => [4,2,2] => 8
[3,2,2,1] => [4,3,1] => 9
[3,2,1,1,1] => [5,2,1] => 11
[3,1,1,1,1,1] => [6,1,1] => 15
[2,2,2,2] => [4,4] => 12
[2,2,2,1,1] => [5,3] => 13
[2,2,1,1,1,1] => [6,2] => 16
[2,1,1,1,1,1,1] => [7,1] => 21
[1,1,1,1,1,1,1,1] => [8] => 28
[9] => [1,1,1,1,1,1,1,1,1] => 0
[8,1] => [2,1,1,1,1,1,1,1] => 1
[7,2] => [2,2,1,1,1,1,1] => 2
[7,1,1] => [3,1,1,1,1,1,1] => 3
[6,3] => [2,2,2,1,1,1] => 3
[6,2,1] => [3,2,1,1,1,1] => 4
[6,1,1,1] => [4,1,1,1,1,1] => 6
[5,4] => [2,2,2,2,1] => 4
[5,3,1] => [3,2,2,1,1] => 5
[5,2,2] => [3,3,1,1,1] => 6
[5,2,1,1] => [4,2,1,1,1] => 7
[5,1,1,1,1] => [5,1,1,1,1] => 10
[4,4,1] => [3,2,2,2] => 6
[4,3,2] => [3,3,2,1] => 7
[4,3,1,1] => [4,2,2,1] => 8
[4,2,2,1] => [4,3,1,1] => 9
[4,2,1,1,1] => [5,2,1,1] => 11
[4,1,1,1,1,1] => [6,1,1,1] => 15
[3,3,3] => [3,3,3] => 9
[3,3,2,1] => [4,3,2] => 10
[3,3,1,1,1] => [5,2,2] => 12
[3,2,2,2] => [4,4,1] => 12
[3,2,2,1,1] => [5,3,1] => 13
[3,2,1,1,1,1] => [6,2,1] => 16
[3,1,1,1,1,1,1] => [7,1,1] => 21
[2,2,2,2,1] => [5,4] => 16
[2,2,2,1,1,1] => [6,3] => 18
[2,2,1,1,1,1,1] => [7,2] => 22
[2,1,1,1,1,1,1,1] => [8,1] => 28
[1,1,1,1,1,1,1,1,1] => [9] => 36
[10] => [1,1,1,1,1,1,1,1,1,1] => 0
[9,1] => [2,1,1,1,1,1,1,1,1] => 1
[8,2] => [2,2,1,1,1,1,1,1] => 2
[8,1,1] => [3,1,1,1,1,1,1,1] => 3
[7,3] => [2,2,2,1,1,1,1] => 3
[7,2,1] => [3,2,1,1,1,1,1] => 4
>>> Load all 270 entries. <<<
[7,1,1,1] => [4,1,1,1,1,1,1] => 6
[6,4] => [2,2,2,2,1,1] => 4
[6,3,1] => [3,2,2,1,1,1] => 5
[6,2,2] => [3,3,1,1,1,1] => 6
[6,2,1,1] => [4,2,1,1,1,1] => 7
[6,1,1,1,1] => [5,1,1,1,1,1] => 10
[5,5] => [2,2,2,2,2] => 5
[5,4,1] => [3,2,2,2,1] => 6
[5,3,2] => [3,3,2,1,1] => 7
[5,3,1,1] => [4,2,2,1,1] => 8
[5,2,2,1] => [4,3,1,1,1] => 9
[5,2,1,1,1] => [5,2,1,1,1] => 11
[5,1,1,1,1,1] => [6,1,1,1,1] => 15
[4,4,2] => [3,3,2,2] => 8
[4,4,1,1] => [4,2,2,2] => 9
[4,3,3] => [3,3,3,1] => 9
[4,3,2,1] => [4,3,2,1] => 10
[4,3,1,1,1] => [5,2,2,1] => 12
[4,2,2,2] => [4,4,1,1] => 12
[4,2,2,1,1] => [5,3,1,1] => 13
[4,2,1,1,1,1] => [6,2,1,1] => 16
[4,1,1,1,1,1,1] => [7,1,1,1] => 21
[3,3,3,1] => [4,3,3] => 12
[3,3,2,2] => [4,4,2] => 13
[3,3,2,1,1] => [5,3,2] => 14
[3,3,1,1,1,1] => [6,2,2] => 17
[3,2,2,2,1] => [5,4,1] => 16
[3,2,2,1,1,1] => [6,3,1] => 18
[3,2,1,1,1,1,1] => [7,2,1] => 22
[3,1,1,1,1,1,1,1] => [8,1,1] => 28
[2,2,2,2,2] => [5,5] => 20
[2,2,2,2,1,1] => [6,4] => 21
[2,2,2,1,1,1,1] => [7,3] => 24
[2,2,1,1,1,1,1,1] => [8,2] => 29
[2,1,1,1,1,1,1,1,1] => [9,1] => 36
[1,1,1,1,1,1,1,1,1,1] => [10] => 45
[11] => [1,1,1,1,1,1,1,1,1,1,1] => 0
[10,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[9,2] => [2,2,1,1,1,1,1,1,1] => 2
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 3
[8,3] => [2,2,2,1,1,1,1,1] => 3
[8,2,1] => [3,2,1,1,1,1,1,1] => 4
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 6
[7,4] => [2,2,2,2,1,1,1] => 4
[7,3,1] => [3,2,2,1,1,1,1] => 5
[7,2,2] => [3,3,1,1,1,1,1] => 6
[7,2,1,1] => [4,2,1,1,1,1,1] => 7
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 10
[6,5] => [2,2,2,2,2,1] => 5
[6,4,1] => [3,2,2,2,1,1] => 6
[6,3,2] => [3,3,2,1,1,1] => 7
[6,3,1,1] => [4,2,2,1,1,1] => 8
[6,2,2,1] => [4,3,1,1,1,1] => 9
[6,2,1,1,1] => [5,2,1,1,1,1] => 11
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 15
[5,5,1] => [3,2,2,2,2] => 7
[5,4,2] => [3,3,2,2,1] => 8
[5,4,1,1] => [4,2,2,2,1] => 9
[5,3,3] => [3,3,3,1,1] => 9
[5,3,2,1] => [4,3,2,1,1] => 10
[5,3,1,1,1] => [5,2,2,1,1] => 12
[5,2,2,2] => [4,4,1,1,1] => 12
[5,2,2,1,1] => [5,3,1,1,1] => 13
[5,2,1,1,1,1] => [6,2,1,1,1] => 16
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 21
[4,4,3] => [3,3,3,2] => 10
[4,4,2,1] => [4,3,2,2] => 11
[4,4,1,1,1] => [5,2,2,2] => 13
[4,3,3,1] => [4,3,3,1] => 12
[4,3,2,2] => [4,4,2,1] => 13
[4,3,2,1,1] => [5,3,2,1] => 14
[4,3,1,1,1,1] => [6,2,2,1] => 17
[4,2,2,2,1] => [5,4,1,1] => 16
[4,2,2,1,1,1] => [6,3,1,1] => 18
[4,2,1,1,1,1,1] => [7,2,1,1] => 22
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 28
[3,3,3,2] => [4,4,3] => 15
[3,3,3,1,1] => [5,3,3] => 16
[3,3,2,2,1] => [5,4,2] => 17
[3,3,2,1,1,1] => [6,3,2] => 19
[3,3,1,1,1,1,1] => [7,2,2] => 23
[3,2,2,2,2] => [5,5,1] => 20
[3,2,2,2,1,1] => [6,4,1] => 21
[3,2,2,1,1,1,1] => [7,3,1] => 24
[3,2,1,1,1,1,1,1] => [8,2,1] => 29
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 36
[2,2,2,2,2,1] => [6,5] => 25
[2,2,2,2,1,1,1] => [7,4] => 27
[2,2,2,1,1,1,1,1] => [8,3] => 31
[2,2,1,1,1,1,1,1,1] => [9,2] => 37
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 45
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 55
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 1
[10,2] => [2,2,1,1,1,1,1,1,1,1] => 2
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 3
[9,3] => [2,2,2,1,1,1,1,1,1] => 3
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 4
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 6
[8,4] => [2,2,2,2,1,1,1,1] => 4
[8,3,1] => [3,2,2,1,1,1,1,1] => 5
[8,2,2] => [3,3,1,1,1,1,1,1] => 6
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 7
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 10
[7,5] => [2,2,2,2,2,1,1] => 5
[7,4,1] => [3,2,2,2,1,1,1] => 6
[7,3,2] => [3,3,2,1,1,1,1] => 7
[7,3,1,1] => [4,2,2,1,1,1,1] => 8
[7,2,2,1] => [4,3,1,1,1,1,1] => 9
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 11
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 15
[6,6] => [2,2,2,2,2,2] => 6
[6,5,1] => [3,2,2,2,2,1] => 7
[6,4,2] => [3,3,2,2,1,1] => 8
[6,4,1,1] => [4,2,2,2,1,1] => 9
[6,3,3] => [3,3,3,1,1,1] => 9
[6,3,2,1] => [4,3,2,1,1,1] => 10
[6,3,1,1,1] => [5,2,2,1,1,1] => 12
[6,2,2,2] => [4,4,1,1,1,1] => 12
[6,2,2,1,1] => [5,3,1,1,1,1] => 13
[6,2,1,1,1,1] => [6,2,1,1,1,1] => 16
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => 21
[5,5,2] => [3,3,2,2,2] => 9
[5,5,1,1] => [4,2,2,2,2] => 10
[5,4,3] => [3,3,3,2,1] => 10
[5,4,2,1] => [4,3,2,2,1] => 11
[5,4,1,1,1] => [5,2,2,2,1] => 13
[5,3,3,1] => [4,3,3,1,1] => 12
[5,3,2,2] => [4,4,2,1,1] => 13
[5,3,2,1,1] => [5,3,2,1,1] => 14
[5,3,1,1,1,1] => [6,2,2,1,1] => 17
[5,2,2,2,1] => [5,4,1,1,1] => 16
[5,2,2,1,1,1] => [6,3,1,1,1] => 18
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 22
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 28
[4,4,4] => [3,3,3,3] => 12
[4,4,3,1] => [4,3,3,2] => 13
[4,4,2,2] => [4,4,2,2] => 14
[4,4,2,1,1] => [5,3,2,2] => 15
[4,4,1,1,1,1] => [6,2,2,2] => 18
[4,3,3,2] => [4,4,3,1] => 15
[4,3,3,1,1] => [5,3,3,1] => 16
[4,3,2,2,1] => [5,4,2,1] => 17
[4,3,2,1,1,1] => [6,3,2,1] => 19
[4,3,1,1,1,1,1] => [7,2,2,1] => 23
[4,2,2,2,2] => [5,5,1,1] => 20
[4,2,2,2,1,1] => [6,4,1,1] => 21
[4,2,2,1,1,1,1] => [7,3,1,1] => 24
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 29
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 36
[3,3,3,3] => [4,4,4] => 18
[3,3,3,2,1] => [5,4,3] => 19
[3,3,3,1,1,1] => [6,3,3] => 21
[3,3,2,2,2] => [5,5,2] => 21
[3,3,2,2,1,1] => [6,4,2] => 22
[3,3,2,1,1,1,1] => [7,3,2] => 25
[3,3,1,1,1,1,1,1] => [8,2,2] => 30
[3,2,2,2,2,1] => [6,5,1] => 25
[3,2,2,2,1,1,1] => [7,4,1] => 27
[3,2,2,1,1,1,1,1] => [8,3,1] => 31
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 37
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 45
[2,2,2,2,2,2] => [6,6] => 30
[2,2,2,2,2,1,1] => [7,5] => 31
[2,2,2,2,1,1,1,1] => [8,4] => 34
[2,2,2,1,1,1,1,1,1] => [9,3] => 39
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 46
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 55
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 66
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Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
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.