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-----------------------------------------------------------------------------
Statistic identifier: St000567
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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].
-----------------------------------------------------------------------------
References: [1] Kopitzke, G. The Gini Index of an Integer Partition [[arXiv:2005.04284]]
[2] Hohlweg, C. Minimal and maximal elements in Kazhdan-Lusztig double sided cells of $S_n$ and Robinson-Schensted correspondance [[arXiv:math/0304059]]
-----------------------------------------------------------------------------
Code:
def statistic(L):
return sum(L[a]*L[b] for a,b in Subsets(range(len(L)), 2))
-----------------------------------------------------------------------------
Statistic values:
[2] => 0
[1,1] => 1
[3] => 0
[2,1] => 2
[1,1,1] => 3
[4] => 0
[3,1] => 3
[2,2] => 4
[2,1,1] => 5
[1,1,1,1] => 6
[5] => 0
[4,1] => 4
[3,2] => 6
[3,1,1] => 7
[2,2,1] => 8
[2,1,1,1] => 9
[1,1,1,1,1] => 10
[6] => 0
[5,1] => 5
[4,2] => 8
[4,1,1] => 9
[3,3] => 9
[3,2,1] => 11
[3,1,1,1] => 12
[2,2,2] => 12
[2,2,1,1] => 13
[2,1,1,1,1] => 14
[1,1,1,1,1,1] => 15
[7] => 0
[6,1] => 6
[5,2] => 10
[5,1,1] => 11
[4,3] => 12
[4,2,1] => 14
[4,1,1,1] => 15
[3,3,1] => 15
[3,2,2] => 16
[3,2,1,1] => 17
[3,1,1,1,1] => 18
[2,2,2,1] => 18
[2,2,1,1,1] => 19
[2,1,1,1,1,1] => 20
[1,1,1,1,1,1,1] => 21
[8] => 0
[7,1] => 7
[6,2] => 12
[6,1,1] => 13
[5,3] => 15
[5,2,1] => 17
[5,1,1,1] => 18
[4,4] => 16
[4,3,1] => 19
[4,2,2] => 20
[4,2,1,1] => 21
[4,1,1,1,1] => 22
[3,3,2] => 21
[3,3,1,1] => 22
[3,2,2,1] => 23
[3,2,1,1,1] => 24
[3,1,1,1,1,1] => 25
[2,2,2,2] => 24
[2,2,2,1,1] => 25
[2,2,1,1,1,1] => 26
[2,1,1,1,1,1,1] => 27
[1,1,1,1,1,1,1,1] => 28
[9] => 0
[8,1] => 8
[7,2] => 14
[7,1,1] => 15
[6,3] => 18
[6,2,1] => 20
[6,1,1,1] => 21
[5,4] => 20
[5,3,1] => 23
[5,2,2] => 24
[5,2,1,1] => 25
[5,1,1,1,1] => 26
[4,4,1] => 24
[4,3,2] => 26
[4,3,1,1] => 27
[4,2,2,1] => 28
[4,2,1,1,1] => 29
[4,1,1,1,1,1] => 30
[3,3,3] => 27
[3,3,2,1] => 29
[3,3,1,1,1] => 30
[3,2,2,2] => 30
[3,2,2,1,1] => 31
[3,2,1,1,1,1] => 32
[3,1,1,1,1,1,1] => 33
[2,2,2,2,1] => 32
[2,2,2,1,1,1] => 33
[2,2,1,1,1,1,1] => 34
[2,1,1,1,1,1,1,1] => 35
[1,1,1,1,1,1,1,1,1] => 36
[10] => 0
[9,1] => 9
[8,2] => 16
[8,1,1] => 17
[7,3] => 21
[7,2,1] => 23
[7,1,1,1] => 24
[6,4] => 24
[6,3,1] => 27
[6,2,2] => 28
[6,2,1,1] => 29
[6,1,1,1,1] => 30
[5,5] => 25
[5,4,1] => 29
[5,3,2] => 31
[5,3,1,1] => 32
[5,2,2,1] => 33
[5,2,1,1,1] => 34
[5,1,1,1,1,1] => 35
[4,4,2] => 32
[4,4,1,1] => 33
[4,3,3] => 33
[4,3,2,1] => 35
[4,3,1,1,1] => 36
[4,2,2,2] => 36
[4,2,2,1,1] => 37
[4,2,1,1,1,1] => 38
[4,1,1,1,1,1,1] => 39
[3,3,3,1] => 36
[3,3,2,2] => 37
[3,3,2,1,1] => 38
[3,3,1,1,1,1] => 39
[3,2,2,2,1] => 39
[3,2,2,1,1,1] => 40
[3,2,1,1,1,1,1] => 41
[3,1,1,1,1,1,1,1] => 42
[2,2,2,2,2] => 40
[2,2,2,2,1,1] => 41
[2,2,2,1,1,1,1] => 42
[2,2,1,1,1,1,1,1] => 43
[2,1,1,1,1,1,1,1,1] => 44
[1,1,1,1,1,1,1,1,1,1] => 45
[11] => 0
[10,1] => 10
[9,2] => 18
[9,1,1] => 19
[8,3] => 24
[8,2,1] => 26
[8,1,1,1] => 27
[7,4] => 28
[7,3,1] => 31
[7,2,2] => 32
[7,2,1,1] => 33
[7,1,1,1,1] => 34
[6,5] => 30
[6,4,1] => 34
[6,3,2] => 36
[6,3,1,1] => 37
[6,2,2,1] => 38
[6,2,1,1,1] => 39
[6,1,1,1,1,1] => 40
[5,5,1] => 35
[5,4,2] => 38
[5,4,1,1] => 39
[5,3,3] => 39
[5,3,2,1] => 41
[5,3,1,1,1] => 42
[5,2,2,2] => 42
[5,2,2,1,1] => 43
[5,2,1,1,1,1] => 44
[5,1,1,1,1,1,1] => 45
[4,4,3] => 40
[4,4,2,1] => 42
[4,4,1,1,1] => 43
[4,3,3,1] => 43
[4,3,2,2] => 44
[4,3,2,1,1] => 45
[4,3,1,1,1,1] => 46
[4,2,2,2,1] => 46
[4,2,2,1,1,1] => 47
[4,2,1,1,1,1,1] => 48
[4,1,1,1,1,1,1,1] => 49
[3,3,3,2] => 45
[3,3,3,1,1] => 46
[3,3,2,2,1] => 47
[3,3,2,1,1,1] => 48
[3,3,1,1,1,1,1] => 49
[3,2,2,2,2] => 48
[3,2,2,2,1,1] => 49
[3,2,2,1,1,1,1] => 50
[3,2,1,1,1,1,1,1] => 51
[3,1,1,1,1,1,1,1,1] => 52
[2,2,2,2,2,1] => 50
[2,2,2,2,1,1,1] => 51
[2,2,2,1,1,1,1,1] => 52
[2,2,1,1,1,1,1,1,1] => 53
[2,1,1,1,1,1,1,1,1,1] => 54
[1,1,1,1,1,1,1,1,1,1,1] => 55
[12] => 0
[11,1] => 11
[10,2] => 20
[10,1,1] => 21
[9,3] => 27
[9,2,1] => 29
[9,1,1,1] => 30
[8,4] => 32
[8,3,1] => 35
[8,2,2] => 36
[8,2,1,1] => 37
[8,1,1,1,1] => 38
[7,5] => 35
[7,4,1] => 39
[7,3,2] => 41
[7,3,1,1] => 42
[7,2,2,1] => 43
[7,2,1,1,1] => 44
[7,1,1,1,1,1] => 45
[6,6] => 36
[6,5,1] => 41
[6,4,2] => 44
[6,4,1,1] => 45
[6,3,3] => 45
[6,3,2,1] => 47
[6,3,1,1,1] => 48
[6,2,2,2] => 48
[6,2,2,1,1] => 49
[6,2,1,1,1,1] => 50
[6,1,1,1,1,1,1] => 51
[5,5,2] => 45
[5,5,1,1] => 46
[5,4,3] => 47
[5,4,2,1] => 49
[5,4,1,1,1] => 50
[5,3,3,1] => 50
[5,3,2,2] => 51
[5,3,2,1,1] => 52
[5,3,1,1,1,1] => 53
[5,2,2,2,1] => 53
[5,2,2,1,1,1] => 54
[5,2,1,1,1,1,1] => 55
[5,1,1,1,1,1,1,1] => 56
[4,4,4] => 48
[4,4,3,1] => 51
[4,4,2,2] => 52
[4,4,2,1,1] => 53
[4,4,1,1,1,1] => 54
[4,3,3,2] => 53
[4,3,3,1,1] => 54
[4,3,2,2,1] => 55
[4,3,2,1,1,1] => 56
[4,3,1,1,1,1,1] => 57
[4,2,2,2,2] => 56
[4,2,2,2,1,1] => 57
[4,2,2,1,1,1,1] => 58
[4,2,1,1,1,1,1,1] => 59
[4,1,1,1,1,1,1,1,1] => 60
[3,3,3,3] => 54
[3,3,3,2,1] => 56
[3,3,3,1,1,1] => 57
[3,3,2,2,2] => 57
[3,3,2,2,1,1] => 58
[3,3,2,1,1,1,1] => 59
[3,3,1,1,1,1,1,1] => 60
[3,2,2,2,2,1] => 59
[3,2,2,2,1,1,1] => 60
[3,2,2,1,1,1,1,1] => 61
[3,2,1,1,1,1,1,1,1] => 62
[3,1,1,1,1,1,1,1,1,1] => 63
[2,2,2,2,2,2] => 60
[2,2,2,2,2,1,1] => 61
[2,2,2,2,1,1,1,1] => 62
[2,2,2,1,1,1,1,1,1] => 63
[2,2,1,1,1,1,1,1,1,1] => 64
[2,1,1,1,1,1,1,1,1,1,1] => 65
[1,1,1,1,1,1,1,1,1,1,1,1] => 66
-----------------------------------------------------------------------------
Created: Aug 07, 2016 at 13:18 by Martin Rubey
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Last Updated: Nov 09, 2021 at 15:38 by Martin Rubey