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Identifier

St001541: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 1

[1,1] => 0

[3] => 3

[2,1] => 2

[1,1,1] => 0

[4] => 6

[3,1] => 5

[2,2] => 4

[2,1,1] => 3

[1,1,1,1] => 0

[5] => 10

[4,1] => 9

[3,2] => 8

[3,1,1] => 7

[2,2,1] => 6

[2,1,1,1] => 4

[1,1,1,1,1] => 0

[6] => 15

[5,1] => 14

[4,2] => 13

[4,1,1] => 12

[3,3] => 12

[3,2,1] => 11

[3,1,1,1] => 9

[2,2,2] => 9

[2,2,1,1] => 8

[2,1,1,1,1] => 5

[1,1,1,1,1,1] => 0

[7] => 21

[6,1] => 20

[5,2] => 19

[5,1,1] => 18

[4,3] => 18

[4,2,1] => 17

[4,1,1,1] => 15

[3,3,1] => 16

[3,2,2] => 15

[3,2,1,1] => 14

[3,1,1,1,1] => 11

[2,2,2,1] => 12

[2,2,1,1,1] => 10

[2,1,1,1,1,1] => 6

[1,1,1,1,1,1,1] => 0

[8] => 28

[7,1] => 27

[6,2] => 26

[6,1,1] => 25

[5,3] => 25

[5,2,1] => 24

[5,1,1,1] => 22

[4,4] => 24

[4,3,1] => 23

[4,2,2] => 22

[4,2,1,1] => 21

[4,1,1,1,1] => 18

[3,3,2] => 21

[3,3,1,1] => 20

[3,2,2,1] => 19

[3,2,1,1,1] => 17

[3,1,1,1,1,1] => 13

[2,2,2,2] => 16

[2,2,2,1,1] => 15

[2,2,1,1,1,1] => 12

[2,1,1,1,1,1,1] => 7

[1,1,1,1,1,1,1,1] => 0

[9] => 36

[8,1] => 35

[7,2] => 34

[7,1,1] => 33

[6,3] => 33

[6,2,1] => 32

[6,1,1,1] => 30

[5,4] => 32

[5,3,1] => 31

[5,2,2] => 30

[5,2,1,1] => 29

[5,1,1,1,1] => 26

[4,4,1] => 30

[4,3,2] => 29

[4,3,1,1] => 28

[4,2,2,1] => 27

[4,2,1,1,1] => 25

[4,1,1,1,1,1] => 21

[3,3,3] => 27

[3,3,2,1] => 26

[3,3,1,1,1] => 24

[3,2,2,2] => 24

[3,2,2,1,1] => 23

[3,2,1,1,1,1] => 20

[3,1,1,1,1,1,1] => 15

[2,2,2,2,1] => 20

[2,2,2,1,1,1] => 18

[2,2,1,1,1,1,1] => 14

[2,1,1,1,1,1,1,1] => 8

[1,1,1,1,1,1,1,1,1] => 0

[10] => 45

[9,1] => 44

[8,2] => 43

[8,1,1] => 42

[7,3] => 42

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Description

The Gini index of an integer partition.
As discussed in [1], this statistic is equal to St000567The sum of the products of all pairs of parts. applied to the conjugate partition.

References

[1] Kopitzke, G. The Gini Index of an Integer Partition arXiv:2005.04284

Code

def statistic(L):
    L = L.conjugate()
    n = sum(L)
    return binomial(n+1,2) - sum(binomial(l+1,2) for l in L)

Created

May 12, 2020 at 15:22 by Christian Stump

Updated

May 12, 2020 at 15:22 by Christian Stump