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Identifier

St001178: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 0

[1,1] => 0

[3] => 0

[2,1] => 3

[1,1,1] => 0

[4] => 0

[3,1] => 8

[2,2] => 12

[2,1,1] => 8

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

[5] => 0

[4,1] => 15

[3,2] => 24

[3,1,1] => 20

[2,2,1] => 24

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

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

[6] => 0

[5,1] => 24

[4,2] => 40

[4,1,1] => 36

[3,3] => 48

[3,2,1] => 45

[3,1,1,1] => 36

[2,2,2] => 48

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

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

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

[7] => 0

[6,1] => 35

[5,2] => 60

[5,1,1] => 56

[4,3] => 75

[4,2,1] => 72

[4,1,1,1] => 63

[3,3,1] => 80

[3,2,2] => 80

[3,2,1,1] => 72

[3,1,1,1,1] => 56

[2,2,2,1] => 75

[2,2,1,1,1] => 60

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

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

[8] => 0

[7,1] => 48

[6,2] => 84

[6,1,1] => 80

[5,3] => 108

[5,2,1] => 105

[5,1,1,1] => 96

[4,4] => 120

[4,3,1] => 120

[4,2,2] => 120

[4,2,1,1] => 112

[4,1,1,1,1] => 96

[3,3,2] => 128

[3,3,1,1] => 120

[3,2,2,1] => 120

[3,2,1,1,1] => 105

[3,1,1,1,1,1] => 80

[2,2,2,2] => 120

[2,2,2,1,1] => 108

[2,2,1,1,1,1] => 84

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

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

[9] => 0

[8,1] => 63

[7,2] => 112

[7,1,1] => 108

[6,3] => 147

[6,2,1] => 144

[6,1,1,1] => 135

[5,4] => 168

[5,3,1] => 168

[5,2,2] => 168

[5,2,1,1] => 160

[5,1,1,1,1] => 144

[4,4,1] => 180

[4,3,2] => 183

[4,3,1,1] => 175

[4,2,2,1] => 175

[4,2,1,1,1] => 160

[4,1,1,1,1,1] => 135

[3,3,3] => 192

[3,3,2,1] => 183

[3,3,1,1,1] => 168

[3,2,2,2] => 180

[3,2,2,1,1] => 168

[3,2,1,1,1,1] => 144

[3,1,1,1,1,1,1] => 108

[2,2,2,2,1] => 168

[2,2,2,1,1,1] => 147

[2,2,1,1,1,1,1] => 112

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

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

[10] => 0

[9,1] => 80

[8,2] => 144

[8,1,1] => 140

[7,3] => 192

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Description

Twelve times the variance of the major index among all standard Young tableaux of a partition.
For a partition $\lambda$ of $n$, this variance is given in [1, Proposition 3.2] by
$$\frac{1}{12}\Big(\sum_{k = 1}^n i^2 - \sum_{i,j \in \lambda} h_{ij}^2\Big),$$
where the second sum ranges over all cells in $\lambda$ and $h_{ij}$ is the hook length of the cell $(i,j) \in \lambda$.

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 sum(i^2 for i in [1 .. n] - sum(i^2 for h in L.hook_lengths() for i in h)

Created

May 07, 2018 at 20:47 by Christian Stump

Updated

May 07, 2018 at 20:47 by Christian Stump