- FindStat

Identifier

St000566: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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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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
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

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

Code

def statistic(pi):
    return sum(binomial(p, Integer(2)) for p in pi)

Created

Aug 07, 2016 at 13:27 by Martin Rubey

Updated

Sep 07, 2024 at 01:41 by Sara Billey