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Statistic identifier: St000185
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Collection: Integer partitions
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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].
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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]]
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Code:
def statistic(L):
return L.weighted_size()
-----------------------------------------------------------------------------
Statistic values:
[] => 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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Created: May 07, 2014 at 03:07 by Lahiru Kariyawasam
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Last Updated: Oct 11, 2023 at 15:39 by Martin Rubey