Identifier
Values
=>
Cc0002;cc-rep
[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
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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].
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
[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))
def statistic_alt(L):
n = sum(L)
return binomial(n+1,2) - sum(binomial(l+1,2) for l in L)
Created
Aug 07, 2016 at 13:18 by Martin Rubey
Updated
Nov 09, 2021 at 15:38 by Martin Rubey
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