Identifier
Values
[2] => [1,1] => 1
[1,1] => [2] => 0
[3] => [1,1,1] => 3
[2,1] => [2,1] => 2
[1,1,1] => [3] => 0
[4] => [1,1,1,1] => 6
[3,1] => [2,1,1] => 5
[2,2] => [2,2] => 4
[2,1,1] => [3,1] => 3
[1,1,1,1] => [4] => 0
[5] => [1,1,1,1,1] => 10
[4,1] => [2,1,1,1] => 9
[3,2] => [2,2,1] => 8
[3,1,1] => [3,1,1] => 7
[2,2,1] => [3,2] => 6
[2,1,1,1] => [4,1] => 4
[1,1,1,1,1] => [5] => 0
[6] => [1,1,1,1,1,1] => 15
[5,1] => [2,1,1,1,1] => 14
[4,2] => [2,2,1,1] => 13
[4,1,1] => [3,1,1,1] => 12
[3,3] => [2,2,2] => 12
[3,2,1] => [3,2,1] => 11
[3,1,1,1] => [4,1,1] => 9
[2,2,2] => [3,3] => 9
[2,2,1,1] => [4,2] => 8
[2,1,1,1,1] => [5,1] => 5
[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] => 20
[5,2] => [2,2,1,1,1] => 19
[5,1,1] => [3,1,1,1,1] => 18
[4,3] => [2,2,2,1] => 18
[4,2,1] => [3,2,1,1] => 17
[4,1,1,1] => [4,1,1,1] => 15
[3,3,1] => [3,2,2] => 16
[3,2,2] => [3,3,1] => 15
[3,2,1,1] => [4,2,1] => 14
[3,1,1,1,1] => [5,1,1] => 11
[2,2,2,1] => [4,3] => 12
[2,2,1,1,1] => [5,2] => 10
[2,1,1,1,1,1] => [6,1] => 6
[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] => 27
[6,2] => [2,2,1,1,1,1] => 26
[6,1,1] => [3,1,1,1,1,1] => 25
[5,3] => [2,2,2,1,1] => 25
[5,2,1] => [3,2,1,1,1] => 24
[5,1,1,1] => [4,1,1,1,1] => 22
[4,4] => [2,2,2,2] => 24
[4,3,1] => [3,2,2,1] => 23
[4,2,2] => [3,3,1,1] => 22
[4,2,1,1] => [4,2,1,1] => 21
[4,1,1,1,1] => [5,1,1,1] => 18
[3,3,2] => [3,3,2] => 21
[3,3,1,1] => [4,2,2] => 20
[3,2,2,1] => [4,3,1] => 19
[3,2,1,1,1] => [5,2,1] => 17
[3,1,1,1,1,1] => [6,1,1] => 13
[2,2,2,2] => [4,4] => 16
[2,2,2,1,1] => [5,3] => 15
[2,2,1,1,1,1] => [6,2] => 12
[2,1,1,1,1,1,1] => [7,1] => 7
[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] => 35
[7,2] => [2,2,1,1,1,1,1] => 34
[7,1,1] => [3,1,1,1,1,1,1] => 33
[6,3] => [2,2,2,1,1,1] => 33
[6,2,1] => [3,2,1,1,1,1] => 32
[6,1,1,1] => [4,1,1,1,1,1] => 30
[5,4] => [2,2,2,2,1] => 32
[5,3,1] => [3,2,2,1,1] => 31
[5,2,2] => [3,3,1,1,1] => 30
[5,2,1,1] => [4,2,1,1,1] => 29
[5,1,1,1,1] => [5,1,1,1,1] => 26
[4,4,1] => [3,2,2,2] => 30
[4,3,2] => [3,3,2,1] => 29
[4,3,1,1] => [4,2,2,1] => 28
[4,2,2,1] => [4,3,1,1] => 27
[4,2,1,1,1] => [5,2,1,1] => 25
[4,1,1,1,1,1] => [6,1,1,1] => 21
[3,3,3] => [3,3,3] => 27
[3,3,2,1] => [4,3,2] => 26
[3,3,1,1,1] => [5,2,2] => 24
[3,2,2,2] => [4,4,1] => 24
[3,2,2,1,1] => [5,3,1] => 23
[3,2,1,1,1,1] => [6,2,1] => 20
[3,1,1,1,1,1,1] => [7,1,1] => 15
[2,2,2,2,1] => [5,4] => 20
[2,2,2,1,1,1] => [6,3] => 18
[2,2,1,1,1,1,1] => [7,2] => 14
[2,1,1,1,1,1,1,1] => [8,1] => 8
[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] => 44
[8,2] => [2,2,1,1,1,1,1,1] => 43
[8,1,1] => [3,1,1,1,1,1,1,1] => 42
[7,3] => [2,2,2,1,1,1,1] => 42
[7,2,1] => [3,2,1,1,1,1,1] => 41
>>> Load all 270 entries. <<<
[7,1,1,1] => [4,1,1,1,1,1,1] => 39
[6,4] => [2,2,2,2,1,1] => 41
[6,3,1] => [3,2,2,1,1,1] => 40
[6,2,2] => [3,3,1,1,1,1] => 39
[6,2,1,1] => [4,2,1,1,1,1] => 38
[6,1,1,1,1] => [5,1,1,1,1,1] => 35
[5,5] => [2,2,2,2,2] => 40
[5,4,1] => [3,2,2,2,1] => 39
[5,3,2] => [3,3,2,1,1] => 38
[5,3,1,1] => [4,2,2,1,1] => 37
[5,2,2,1] => [4,3,1,1,1] => 36
[5,2,1,1,1] => [5,2,1,1,1] => 34
[5,1,1,1,1,1] => [6,1,1,1,1] => 30
[4,4,2] => [3,3,2,2] => 37
[4,4,1,1] => [4,2,2,2] => 36
[4,3,3] => [3,3,3,1] => 36
[4,3,2,1] => [4,3,2,1] => 35
[4,3,1,1,1] => [5,2,2,1] => 33
[4,2,2,2] => [4,4,1,1] => 33
[4,2,2,1,1] => [5,3,1,1] => 32
[4,2,1,1,1,1] => [6,2,1,1] => 29
[4,1,1,1,1,1,1] => [7,1,1,1] => 24
[3,3,3,1] => [4,3,3] => 33
[3,3,2,2] => [4,4,2] => 32
[3,3,2,1,1] => [5,3,2] => 31
[3,3,1,1,1,1] => [6,2,2] => 28
[3,2,2,2,1] => [5,4,1] => 29
[3,2,2,1,1,1] => [6,3,1] => 27
[3,2,1,1,1,1,1] => [7,2,1] => 23
[3,1,1,1,1,1,1,1] => [8,1,1] => 17
[2,2,2,2,2] => [5,5] => 25
[2,2,2,2,1,1] => [6,4] => 24
[2,2,2,1,1,1,1] => [7,3] => 21
[2,2,1,1,1,1,1,1] => [8,2] => 16
[2,1,1,1,1,1,1,1,1] => [9,1] => 9
[1,1,1,1,1,1,1,1,1,1] => [10] => 0
[11] => [1,1,1,1,1,1,1,1,1,1,1] => 55
[10,1] => [2,1,1,1,1,1,1,1,1,1] => 54
[9,2] => [2,2,1,1,1,1,1,1,1] => 53
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 52
[8,3] => [2,2,2,1,1,1,1,1] => 52
[8,2,1] => [3,2,1,1,1,1,1,1] => 51
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 49
[7,4] => [2,2,2,2,1,1,1] => 51
[7,3,1] => [3,2,2,1,1,1,1] => 50
[7,2,2] => [3,3,1,1,1,1,1] => 49
[7,2,1,1] => [4,2,1,1,1,1,1] => 48
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 45
[6,5] => [2,2,2,2,2,1] => 50
[6,4,1] => [3,2,2,2,1,1] => 49
[6,3,2] => [3,3,2,1,1,1] => 48
[6,3,1,1] => [4,2,2,1,1,1] => 47
[6,2,2,1] => [4,3,1,1,1,1] => 46
[6,2,1,1,1] => [5,2,1,1,1,1] => 44
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 40
[5,5,1] => [3,2,2,2,2] => 48
[5,4,2] => [3,3,2,2,1] => 47
[5,4,1,1] => [4,2,2,2,1] => 46
[5,3,3] => [3,3,3,1,1] => 46
[5,3,2,1] => [4,3,2,1,1] => 45
[5,3,1,1,1] => [5,2,2,1,1] => 43
[5,2,2,2] => [4,4,1,1,1] => 43
[5,2,2,1,1] => [5,3,1,1,1] => 42
[5,2,1,1,1,1] => [6,2,1,1,1] => 39
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 34
[4,4,3] => [3,3,3,2] => 45
[4,4,2,1] => [4,3,2,2] => 44
[4,4,1,1,1] => [5,2,2,2] => 42
[4,3,3,1] => [4,3,3,1] => 43
[4,3,2,2] => [4,4,2,1] => 42
[4,3,2,1,1] => [5,3,2,1] => 41
[4,3,1,1,1,1] => [6,2,2,1] => 38
[4,2,2,2,1] => [5,4,1,1] => 39
[4,2,2,1,1,1] => [6,3,1,1] => 37
[4,2,1,1,1,1,1] => [7,2,1,1] => 33
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 27
[3,3,3,2] => [4,4,3] => 40
[3,3,3,1,1] => [5,3,3] => 39
[3,3,2,2,1] => [5,4,2] => 38
[3,3,2,1,1,1] => [6,3,2] => 36
[3,3,1,1,1,1,1] => [7,2,2] => 32
[3,2,2,2,2] => [5,5,1] => 35
[3,2,2,2,1,1] => [6,4,1] => 34
[3,2,2,1,1,1,1] => [7,3,1] => 31
[3,2,1,1,1,1,1,1] => [8,2,1] => 26
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 19
[2,2,2,2,2,1] => [6,5] => 30
[2,2,2,2,1,1,1] => [7,4] => 28
[2,2,2,1,1,1,1,1] => [8,3] => 24
[2,2,1,1,1,1,1,1,1] => [9,2] => 18
[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] => 0
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 66
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 65
[10,2] => [2,2,1,1,1,1,1,1,1,1] => 64
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 63
[9,3] => [2,2,2,1,1,1,1,1,1] => 63
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 62
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 60
[8,4] => [2,2,2,2,1,1,1,1] => 62
[8,3,1] => [3,2,2,1,1,1,1,1] => 61
[8,2,2] => [3,3,1,1,1,1,1,1] => 60
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 59
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 56
[7,5] => [2,2,2,2,2,1,1] => 61
[7,4,1] => [3,2,2,2,1,1,1] => 60
[7,3,2] => [3,3,2,1,1,1,1] => 59
[7,3,1,1] => [4,2,2,1,1,1,1] => 58
[7,2,2,1] => [4,3,1,1,1,1,1] => 57
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 55
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 51
[6,6] => [2,2,2,2,2,2] => 60
[6,5,1] => [3,2,2,2,2,1] => 59
[6,4,2] => [3,3,2,2,1,1] => 58
[6,4,1,1] => [4,2,2,2,1,1] => 57
[6,3,3] => [3,3,3,1,1,1] => 57
[6,3,2,1] => [4,3,2,1,1,1] => 56
[6,3,1,1,1] => [5,2,2,1,1,1] => 54
[6,2,2,2] => [4,4,1,1,1,1] => 54
[6,2,2,1,1] => [5,3,1,1,1,1] => 53
[6,2,1,1,1,1] => [6,2,1,1,1,1] => 50
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => 45
[5,5,2] => [3,3,2,2,2] => 57
[5,5,1,1] => [4,2,2,2,2] => 56
[5,4,3] => [3,3,3,2,1] => 56
[5,4,2,1] => [4,3,2,2,1] => 55
[5,4,1,1,1] => [5,2,2,2,1] => 53
[5,3,3,1] => [4,3,3,1,1] => 54
[5,3,2,2] => [4,4,2,1,1] => 53
[5,3,2,1,1] => [5,3,2,1,1] => 52
[5,3,1,1,1,1] => [6,2,2,1,1] => 49
[5,2,2,2,1] => [5,4,1,1,1] => 50
[5,2,2,1,1,1] => [6,3,1,1,1] => 48
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 44
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 38
[4,4,4] => [3,3,3,3] => 54
[4,4,3,1] => [4,3,3,2] => 53
[4,4,2,2] => [4,4,2,2] => 52
[4,4,2,1,1] => [5,3,2,2] => 51
[4,4,1,1,1,1] => [6,2,2,2] => 48
[4,3,3,2] => [4,4,3,1] => 51
[4,3,3,1,1] => [5,3,3,1] => 50
[4,3,2,2,1] => [5,4,2,1] => 49
[4,3,2,1,1,1] => [6,3,2,1] => 47
[4,3,1,1,1,1,1] => [7,2,2,1] => 43
[4,2,2,2,2] => [5,5,1,1] => 46
[4,2,2,2,1,1] => [6,4,1,1] => 45
[4,2,2,1,1,1,1] => [7,3,1,1] => 42
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 37
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 30
[3,3,3,3] => [4,4,4] => 48
[3,3,3,2,1] => [5,4,3] => 47
[3,3,3,1,1,1] => [6,3,3] => 45
[3,3,2,2,2] => [5,5,2] => 45
[3,3,2,2,1,1] => [6,4,2] => 44
[3,3,2,1,1,1,1] => [7,3,2] => 41
[3,3,1,1,1,1,1,1] => [8,2,2] => 36
[3,2,2,2,2,1] => [6,5,1] => 41
[3,2,2,2,1,1,1] => [7,4,1] => 39
[3,2,2,1,1,1,1,1] => [8,3,1] => 35
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 29
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 21
[2,2,2,2,2,2] => [6,6] => 36
[2,2,2,2,2,1,1] => [7,5] => 35
[2,2,2,2,1,1,1,1] => [8,4] => 32
[2,2,2,1,1,1,1,1,1] => [9,3] => 27
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 20
[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] => 0
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Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
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.