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Statistic identifier: St000566
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Collection: Integer partitions
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Description: The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
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References: [1] Xi, N."The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group Sn," , p. 4. Xi, N. The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group $S_n$ [[arXiv:math/0401430]]
[2] Lusztig, G. Cells in affine Weyl groups [[MathSciNet:0803338]]
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Code:
def statistic(pi):
return sum(binomial(p, Integer(2)) for p in pi)
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 0
[3] => 3
[2,1] => 1
[1,1,1] => 0
[4] => 6
[3,1] => 3
[2,2] => 2
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 10
[4,1] => 6
[3,2] => 4
[3,1,1] => 3
[2,2,1] => 2
[2,1,1,1] => 1
[1,1,1,1,1] => 0
[6] => 15
[5,1] => 10
[4,2] => 7
[4,1,1] => 6
[3,3] => 6
[3,2,1] => 4
[3,1,1,1] => 3
[2,2,2] => 3
[2,2,1,1] => 2
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 0
[7] => 21
[6,1] => 15
[5,2] => 11
[5,1,1] => 10
[4,3] => 9
[4,2,1] => 7
[4,1,1,1] => 6
[3,3,1] => 6
[3,2,2] => 5
[3,2,1,1] => 4
[3,1,1,1,1] => 3
[2,2,2,1] => 3
[2,2,1,1,1] => 2
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 0
[8] => 28
[7,1] => 21
[6,2] => 16
[6,1,1] => 15
[5,3] => 13
[5,2,1] => 11
[5,1,1,1] => 10
[4,4] => 12
[4,3,1] => 9
[4,2,2] => 8
[4,2,1,1] => 7
[4,1,1,1,1] => 6
[3,3,2] => 7
[3,3,1,1] => 6
[3,2,2,1] => 5
[3,2,1,1,1] => 4
[3,1,1,1,1,1] => 3
[2,2,2,2] => 4
[2,2,2,1,1] => 3
[2,2,1,1,1,1] => 2
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 0
[9] => 36
[8,1] => 28
[7,2] => 22
[7,1,1] => 21
[6,3] => 18
[6,2,1] => 16
[6,1,1,1] => 15
[5,4] => 16
[5,3,1] => 13
[5,2,2] => 12
[5,2,1,1] => 11
[5,1,1,1,1] => 10
[4,4,1] => 12
[4,3,2] => 10
[4,3,1,1] => 9
[4,2,2,1] => 8
[4,2,1,1,1] => 7
[4,1,1,1,1,1] => 6
[3,3,3] => 9
[3,3,2,1] => 7
[3,3,1,1,1] => 6
[3,2,2,2] => 6
[3,2,2,1,1] => 5
[3,2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => 3
[2,2,2,2,1] => 4
[2,2,2,1,1,1] => 3
[2,2,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 0
[10] => 45
[9,1] => 36
[8,2] => 29
[8,1,1] => 28
[7,3] => 24
[7,2,1] => 22
[7,1,1,1] => 21
[6,4] => 21
[6,3,1] => 18
[6,2,2] => 17
[6,2,1,1] => 16
[6,1,1,1,1] => 15
[5,5] => 20
[5,4,1] => 16
[5,3,2] => 14
[5,3,1,1] => 13
[5,2,2,1] => 12
[5,2,1,1,1] => 11
[5,1,1,1,1,1] => 10
[4,4,2] => 13
[4,4,1,1] => 12
[4,3,3] => 12
[4,3,2,1] => 10
[4,3,1,1,1] => 9
[4,2,2,2] => 9
[4,2,2,1,1] => 8
[4,2,1,1,1,1] => 7
[4,1,1,1,1,1,1] => 6
[3,3,3,1] => 9
[3,3,2,2] => 8
[3,3,2,1,1] => 7
[3,3,1,1,1,1] => 6
[3,2,2,2,1] => 6
[3,2,2,1,1,1] => 5
[3,2,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1] => 3
[2,2,2,2,2] => 5
[2,2,2,2,1,1] => 4
[2,2,2,1,1,1,1] => 3
[2,2,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 55
[10,1] => 45
[9,2] => 37
[9,1,1] => 36
[8,3] => 31
[8,2,1] => 29
[8,1,1,1] => 28
[7,4] => 27
[7,3,1] => 24
[7,2,2] => 23
[7,2,1,1] => 22
[7,1,1,1,1] => 21
[6,5] => 25
[6,4,1] => 21
[6,3,2] => 19
[6,3,1,1] => 18
[6,2,2,1] => 17
[6,2,1,1,1] => 16
[6,1,1,1,1,1] => 15
[5,5,1] => 20
[5,4,2] => 17
[5,4,1,1] => 16
[5,3,3] => 16
[5,3,2,1] => 14
[5,3,1,1,1] => 13
[5,2,2,2] => 13
[5,2,2,1,1] => 12
[5,2,1,1,1,1] => 11
[5,1,1,1,1,1,1] => 10
[4,4,3] => 15
[4,4,2,1] => 13
[4,4,1,1,1] => 12
[4,3,3,1] => 12
[4,3,2,2] => 11
[4,3,2,1,1] => 10
[4,3,1,1,1,1] => 9
[4,2,2,2,1] => 9
[4,2,2,1,1,1] => 8
[4,2,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1] => 6
[3,3,3,2] => 10
[3,3,3,1,1] => 9
[3,3,2,2,1] => 8
[3,3,2,1,1,1] => 7
[3,3,1,1,1,1,1] => 6
[3,2,2,2,2] => 7
[3,2,2,2,1,1] => 6
[3,2,2,1,1,1,1] => 5
[3,2,1,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1,1] => 3
[2,2,2,2,2,1] => 5
[2,2,2,2,1,1,1] => 4
[2,2,2,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 66
[11,1] => 55
[10,2] => 46
[10,1,1] => 45
[9,3] => 39
[9,2,1] => 37
[9,1,1,1] => 36
[8,4] => 34
[8,3,1] => 31
[8,2,2] => 30
[8,2,1,1] => 29
[8,1,1,1,1] => 28
[7,5] => 31
[7,4,1] => 27
[7,3,2] => 25
[7,3,1,1] => 24
[7,2,2,1] => 23
[7,2,1,1,1] => 22
[7,1,1,1,1,1] => 21
[6,6] => 30
[6,5,1] => 25
[6,4,2] => 22
[6,4,1,1] => 21
[6,3,3] => 21
[6,3,2,1] => 19
[6,3,1,1,1] => 18
[6,2,2,2] => 18
[6,2,2,1,1] => 17
[6,2,1,1,1,1] => 16
[6,1,1,1,1,1,1] => 15
[5,5,2] => 21
[5,5,1,1] => 20
[5,4,3] => 19
[5,4,2,1] => 17
[5,4,1,1,1] => 16
[5,3,3,1] => 16
[5,3,2,2] => 15
[5,3,2,1,1] => 14
[5,3,1,1,1,1] => 13
[5,2,2,2,1] => 13
[5,2,2,1,1,1] => 12
[5,2,1,1,1,1,1] => 11
[5,1,1,1,1,1,1,1] => 10
[4,4,4] => 18
[4,4,3,1] => 15
[4,4,2,2] => 14
[4,4,2,1,1] => 13
[4,4,1,1,1,1] => 12
[4,3,3,2] => 13
[4,3,3,1,1] => 12
[4,3,2,2,1] => 11
[4,3,2,1,1,1] => 10
[4,3,1,1,1,1,1] => 9
[4,2,2,2,2] => 10
[4,2,2,2,1,1] => 9
[4,2,2,1,1,1,1] => 8
[4,2,1,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1,1] => 6
[3,3,3,3] => 12
[3,3,3,2,1] => 10
[3,3,3,1,1,1] => 9
[3,3,2,2,2] => 9
[3,3,2,2,1,1] => 8
[3,3,2,1,1,1,1] => 7
[3,3,1,1,1,1,1,1] => 6
[3,2,2,2,2,1] => 7
[3,2,2,2,1,1,1] => 6
[3,2,2,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1] => 3
[2,2,2,2,2,2] => 6
[2,2,2,2,2,1,1] => 5
[2,2,2,2,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
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Created: Aug 07, 2016 at 13:27 by Martin Rubey
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Last Updated: Sep 07, 2024 at 01:41 by Sara Billey