Identifier
Values
=>
Cc0002;cc-rep
[]=>0
[1]=>0
[2]=>0
[1,1]=>1
[3]=>0
[2,1]=>1
[1,1,1]=>3
[4]=>0
[3,1]=>1
[2,2]=>2
[2,1,1]=>3
[1,1,1,1]=>6
[5]=>0
[4,1]=>1
[3,2]=>2
[3,1,1]=>3
[2,2,1]=>4
[2,1,1,1]=>6
[1,1,1,1,1]=>10
[6]=>0
[5,1]=>1
[4,2]=>2
[4,1,1]=>3
[3,3]=>3
[3,2,1]=>4
[3,1,1,1]=>6
[2,2,2]=>6
[2,2,1,1]=>7
[2,1,1,1,1]=>10
[1,1,1,1,1,1]=>15
[7]=>0
[6,1]=>1
[5,2]=>2
[5,1,1]=>3
[4,3]=>3
[4,2,1]=>4
[4,1,1,1]=>6
[3,3,1]=>5
[3,2,2]=>6
[3,2,1,1]=>7
[3,1,1,1,1]=>10
[2,2,2,1]=>9
[2,2,1,1,1]=>11
[2,1,1,1,1,1]=>15
[1,1,1,1,1,1,1]=>21
[8]=>0
[7,1]=>1
[6,2]=>2
[6,1,1]=>3
[5,3]=>3
[5,2,1]=>4
[5,1,1,1]=>6
[4,4]=>4
[4,3,1]=>5
[4,2,2]=>6
[4,2,1,1]=>7
[4,1,1,1,1]=>10
[3,3,2]=>7
[3,3,1,1]=>8
[3,2,2,1]=>9
[3,2,1,1,1]=>11
[3,1,1,1,1,1]=>15
[2,2,2,2]=>12
[2,2,2,1,1]=>13
[2,2,1,1,1,1]=>16
[2,1,1,1,1,1,1]=>21
[1,1,1,1,1,1,1,1]=>28
[9]=>0
[8,1]=>1
[7,2]=>2
[7,1,1]=>3
[6,3]=>3
[6,2,1]=>4
[6,1,1,1]=>6
[5,4]=>4
[5,3,1]=>5
[5,2,2]=>6
[5,2,1,1]=>7
[5,1,1,1,1]=>10
[4,4,1]=>6
[4,3,2]=>7
[4,3,1,1]=>8
[4,2,2,1]=>9
[4,2,1,1,1]=>11
[4,1,1,1,1,1]=>15
[3,3,3]=>9
[3,3,2,1]=>10
[3,3,1,1,1]=>12
[3,2,2,2]=>12
[3,2,2,1,1]=>13
[3,2,1,1,1,1]=>16
[3,1,1,1,1,1,1]=>21
[2,2,2,2,1]=>16
[2,2,2,1,1,1]=>18
[2,2,1,1,1,1,1]=>22
[2,1,1,1,1,1,1,1]=>28
[1,1,1,1,1,1,1,1,1]=>36
[10]=>0
[9,1]=>1
[8,2]=>2
[8,1,1]=>3
[7,3]=>3
[7,2,1]=>4
[7,1,1,1]=>6
[6,4]=>4
[6,3,1]=>5
[6,2,2]=>6
[6,2,1,1]=>7
[6,1,1,1,1]=>10
[5,5]=>5
[5,4,1]=>6
[5,3,2]=>7
[5,3,1,1]=>8
[5,2,2,1]=>9
[5,2,1,1,1]=>11
[5,1,1,1,1,1]=>15
[4,4,2]=>8
[4,4,1,1]=>9
[4,3,3]=>9
[4,3,2,1]=>10
[4,3,1,1,1]=>12
[4,2,2,2]=>12
[4,2,2,1,1]=>13
[4,2,1,1,1,1]=>16
[4,1,1,1,1,1,1]=>21
[3,3,3,1]=>12
[3,3,2,2]=>13
[3,3,2,1,1]=>14
[3,3,1,1,1,1]=>17
[3,2,2,2,1]=>16
[3,2,2,1,1,1]=>18
[3,2,1,1,1,1,1]=>22
[3,1,1,1,1,1,1,1]=>28
[2,2,2,2,2]=>20
[2,2,2,2,1,1]=>21
[2,2,2,1,1,1,1]=>24
[2,2,1,1,1,1,1,1]=>29
[2,1,1,1,1,1,1,1,1]=>36
[1,1,1,1,1,1,1,1,1,1]=>45
[8,3]=>3
[7,4]=>4
[6,5]=>5
[6,4,1]=>6
[5,5,1]=>7
[5,4,2]=>8
[5,4,1,1]=>9
[5,3,3]=>9
[5,3,2,1]=>10
[5,3,1,1,1]=>12
[5,2,2,2]=>12
[5,2,2,1,1]=>13
[4,4,3]=>10
[4,4,2,1]=>11
[4,4,1,1,1]=>13
[4,3,3,1]=>12
[4,3,2,2]=>13
[4,3,2,1,1]=>14
[4,2,2,2,1]=>16
[3,3,3,2]=>15
[3,3,3,1,1]=>16
[3,3,2,2,1]=>17
[3,2,2,2,2]=>20
[2,2,2,2,2,1]=>25
[7,5]=>5
[7,4,1]=>6
[6,6]=>6
[6,4,2]=>8
[5,5,2]=>9
[5,4,3]=>10
[5,4,2,1]=>11
[5,4,1,1,1]=>13
[5,3,3,1]=>12
[5,3,2,2]=>13
[5,3,2,1,1]=>14
[5,2,2,2,1]=>16
[4,4,4]=>12
[4,4,3,1]=>13
[4,4,2,2]=>14
[4,4,2,1,1]=>15
[4,3,3,2]=>15
[4,3,3,1,1]=>16
[4,3,2,2,1]=>17
[3,3,3,3]=>18
[3,3,3,2,1]=>19
[3,3,2,2,2]=>21
[3,3,2,2,1,1]=>22
[2,2,2,2,2,2]=>30
[8,5]=>5
[7,5,1]=>7
[7,4,2]=>8
[5,5,3]=>11
[5,4,4]=>12
[5,4,3,1]=>13
[5,4,2,2]=>14
[5,4,2,1,1]=>15
[5,3,3,2]=>15
[5,3,3,1,1]=>16
[5,3,2,2,1]=>17
[4,4,4,1]=>15
[4,4,3,2]=>16
[4,4,3,1,1]=>17
[4,4,2,2,1]=>18
[4,3,3,3]=>18
[4,3,3,2,1]=>19
[3,3,3,3,1]=>22
[3,3,3,2,2]=>23
[9,5]=>5
[8,5,1]=>7
[7,5,2]=>9
[7,4,3]=>10
[5,5,4]=>13
[5,4,3,2]=>16
[5,4,3,1,1]=>17
[5,4,2,2,1]=>18
[5,3,3,2,1]=>19
[5,3,2,2,2]=>21
[4,4,4,2]=>18
[4,4,3,3]=>19
[4,4,3,2,1]=>20
[3,3,3,3,2]=>26
[9,5,1]=>7
[8,5,2]=>9
[7,5,3]=>11
[5,5,5]=>15
[5,4,3,2,1]=>20
[5,3,2,2,2,1]=>26
[4,4,4,3]=>21
[3,3,3,3,3]=>30
[8,5,3]=>11
[7,5,3,1]=>14
[4,4,4,4]=>24
[8,6,3]=>12
[9,6,3]=>12
[8,6,4]=>14
[9,6,4]=>14
[8,5,4,2]=>19
[8,5,5,1]=>18
[7,5,4,3,1]=>26
[8,6,4,2]=>20
[10,6,4]=>14
[10,7,3]=>13
[9,7,4]=>15
[9,5,5,1]=>18
[6,5,4,3,2,1]=>35
[11,7,3]=>13
[9,6,4,3]=>23
[9,6,5,3]=>25
[8,6,5,3,1]=>29
[11,7,5,1]=>20
[9,7,5,3]=>26
[9,7,5,3,1]=>30
[10,7,5,3]=>26
[9,7,5,4,1]=>33
[7,6,5,4,3,2,1]=>56
[10,7,6,4,1]=>35
[9,7,6,4,2]=>39
[10,8,5,4,1]=>34
[10,8,6,4,1]=>36
[9,7,5,5,3,1]=>49
[11,8,6,4,1]=>36
[10,8,6,4,2]=>40
[11,8,6,5,1]=>39
[12,9,7,5,1]=>42
[13,9,7,5,1]=>42
[11,9,7,5,3,1]=>55
[11,8,7,5,4,1]=>58
[8,7,6,5,4,3,2,1]=>84
[11,9,7,5,5,3]=>73
[11,9,7,7,5,3,3]=>97
[11,9,7,6,5,3,1]=>82
[13,11,9,7,5,3,1]=>91
[13,11,9,7,7,5,3,1]=>128
[17,13,11,9,7,5,1]=>121
[15,13,11,9,7,5,3,1]=>140
[29,23,19,17,13,11,7,1]=>268
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Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
References
[1] Hohlweg, C. Minimal and maximal elements in Kazhdan-Lusztig double sided cells of $S_n$ and Robinson-Schensted correspondance arXiv:math/0304059
[2] Stanley, R. P. Enumerative combinatorics. Vol. 2 MathSciNet:1676282
[2] Stanley, R. P. Enumerative combinatorics. Vol. 2 MathSciNet:1676282
Code
def statistic(L):
return L.weighted_size()
Created
May 07, 2014 at 03:07 by Lahiru Kariyawasam
Updated
Oct 11, 2023 at 15:39 by Martin Rubey
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