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Identifier

St001177: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 0

[1,1] => 2

[3] => 0

[2,1] => 3

[1,1,1] => 6

[4] => 0

[3,1] => 4

[2,2] => 6

[2,1,1] => 8

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

[5] => 0

[4,1] => 5

[3,2] => 8

[3,1,1] => 10

[2,2,1] => 12

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

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

[6] => 0

[5,1] => 6

[4,2] => 10

[4,1,1] => 12

[3,3] => 12

[3,2,1] => 15

[3,1,1,1] => 18

[2,2,2] => 18

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

[2,1,1,1,1] => 24

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

[7] => 0

[6,1] => 7

[5,2] => 12

[5,1,1] => 14

[4,3] => 15

[4,2,1] => 18

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

[3,3,1] => 20

[3,2,2] => 22

[3,2,1,1] => 24

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

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

[2,2,1,1,1] => 30

[2,1,1,1,1,1] => 35

[1,1,1,1,1,1,1] => 42

[8] => 0

[7,1] => 8

[6,2] => 14

[6,1,1] => 16

[5,3] => 18

[5,2,1] => 21

[5,1,1,1] => 24

[4,4] => 20

[4,3,1] => 24

[4,2,2] => 26

[4,2,1,1] => 28

[4,1,1,1,1] => 32

[3,3,2] => 28

[3,3,1,1] => 30

[3,2,2,1] => 32

[3,2,1,1,1] => 35

[3,1,1,1,1,1] => 40

[2,2,2,2] => 36

[2,2,2,1,1] => 38

[2,2,1,1,1,1] => 42

[2,1,1,1,1,1,1] => 48

[1,1,1,1,1,1,1,1] => 56

[9] => 0

[8,1] => 9

[7,2] => 16

[7,1,1] => 18

[6,3] => 21

[6,2,1] => 24

[6,1,1,1] => 27

[5,4] => 24

[5,3,1] => 28

[5,2,2] => 30

[5,2,1,1] => 32

[5,1,1,1,1] => 36

[4,4,1] => 30

[4,3,2] => 33

[4,3,1,1] => 35

[4,2,2,1] => 37

[4,2,1,1,1] => 40

[4,1,1,1,1,1] => 45

[3,3,3] => 36

[3,3,2,1] => 39

[3,3,1,1,1] => 42

[3,2,2,2] => 42

[3,2,2,1,1] => 44

[3,2,1,1,1,1] => 48

[3,1,1,1,1,1,1] => 54

[2,2,2,2,1] => 48

[2,2,2,1,1,1] => 51

[2,2,1,1,1,1,1] => 56

[2,1,1,1,1,1,1,1] => 63

[1,1,1,1,1,1,1,1,1] => 72

[10] => 0

[9,1] => 10

[8,2] => 18

[8,1,1] => 20

[7,3] => 24

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Description

Twice the mean value of the major index among all standard Young tableaux of a partition.
For a partition $\lambda$ of $n$, this mean value is given in [1, Proposition 3.1] by
$$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$$
where $\lambda_i$ is the size of the $i$-th row of $\lambda$ and $\lambda_i'$ is the size of the $i$-th column.

References

[1] , Adin, R. M., Roichman, Y. Descent functions and random Young tableaux MathSciNet:1841639 arXiv:math/9910165

Code

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

Created

May 07, 2018 at 20:47 by Christian Stump

Updated

May 07, 2018 at 20:47 by Christian Stump