Identifier
- St000055: Permutations ⟶ ℤ (values match St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.)
Values
=>
[1]=>0
[1,2]=>0
[2,1]=>1
[1,2,3]=>0
[1,3,2]=>1
[2,1,3]=>1
[2,3,1]=>3
[3,1,2]=>3
[3,2,1]=>4
[1,2,3,4]=>0
[1,2,4,3]=>1
[1,3,2,4]=>1
[1,3,4,2]=>3
[1,4,2,3]=>3
[1,4,3,2]=>4
[2,1,3,4]=>1
[2,1,4,3]=>2
[2,3,1,4]=>3
[2,3,4,1]=>6
[2,4,1,3]=>5
[2,4,3,1]=>7
[3,1,2,4]=>3
[3,1,4,2]=>5
[3,2,1,4]=>4
[3,2,4,1]=>7
[3,4,1,2]=>8
[3,4,2,1]=>9
[4,1,2,3]=>6
[4,1,3,2]=>7
[4,2,1,3]=>7
[4,2,3,1]=>9
[4,3,1,2]=>9
[4,3,2,1]=>10
[1,2,3,4,5]=>0
[1,2,3,5,4]=>1
[1,2,4,3,5]=>1
[1,2,4,5,3]=>3
[1,2,5,3,4]=>3
[1,2,5,4,3]=>4
[1,3,2,4,5]=>1
[1,3,2,5,4]=>2
[1,3,4,2,5]=>3
[1,3,4,5,2]=>6
[1,3,5,2,4]=>5
[1,3,5,4,2]=>7
[1,4,2,3,5]=>3
[1,4,2,5,3]=>5
[1,4,3,2,5]=>4
[1,4,3,5,2]=>7
[1,4,5,2,3]=>8
[1,4,5,3,2]=>9
[1,5,2,3,4]=>6
[1,5,2,4,3]=>7
[1,5,3,2,4]=>7
[1,5,3,4,2]=>9
[1,5,4,2,3]=>9
[1,5,4,3,2]=>10
[2,1,3,4,5]=>1
[2,1,3,5,4]=>2
[2,1,4,3,5]=>2
[2,1,4,5,3]=>4
[2,1,5,3,4]=>4
[2,1,5,4,3]=>5
[2,3,1,4,5]=>3
[2,3,1,5,4]=>4
[2,3,4,1,5]=>6
[2,3,4,5,1]=>10
[2,3,5,1,4]=>8
[2,3,5,4,1]=>11
[2,4,1,3,5]=>5
[2,4,1,5,3]=>7
[2,4,3,1,5]=>7
[2,4,3,5,1]=>11
[2,4,5,1,3]=>11
[2,4,5,3,1]=>13
[2,5,1,3,4]=>8
[2,5,1,4,3]=>9
[2,5,3,1,4]=>10
[2,5,3,4,1]=>13
[2,5,4,1,3]=>12
[2,5,4,3,1]=>14
[3,1,2,4,5]=>3
[3,1,2,5,4]=>4
[3,1,4,2,5]=>5
[3,1,4,5,2]=>8
[3,1,5,2,4]=>7
[3,1,5,4,2]=>9
[3,2,1,4,5]=>4
[3,2,1,5,4]=>5
[3,2,4,1,5]=>7
[3,2,4,5,1]=>11
[3,2,5,1,4]=>9
[3,2,5,4,1]=>12
[3,4,1,2,5]=>8
[3,4,1,5,2]=>11
[3,4,2,1,5]=>9
[3,4,2,5,1]=>13
[3,4,5,1,2]=>15
[3,4,5,2,1]=>16
[3,5,1,2,4]=>11
[3,5,1,4,2]=>13
[3,5,2,1,4]=>12
[3,5,2,4,1]=>15
[3,5,4,1,2]=>16
[3,5,4,2,1]=>17
[4,1,2,3,5]=>6
[4,1,2,5,3]=>8
[4,1,3,2,5]=>7
[4,1,3,5,2]=>10
[4,1,5,2,3]=>11
[4,1,5,3,2]=>12
[4,2,1,3,5]=>7
[4,2,1,5,3]=>9
[4,2,3,1,5]=>9
[4,2,3,5,1]=>13
[4,2,5,1,3]=>13
[4,2,5,3,1]=>15
[4,3,1,2,5]=>9
[4,3,1,5,2]=>12
[4,3,2,1,5]=>10
[4,3,2,5,1]=>14
[4,3,5,1,2]=>16
[4,3,5,2,1]=>17
[4,5,1,2,3]=>15
[4,5,1,3,2]=>16
[4,5,2,1,3]=>16
[4,5,2,3,1]=>18
[4,5,3,1,2]=>18
[4,5,3,2,1]=>19
[5,1,2,3,4]=>10
[5,1,2,4,3]=>11
[5,1,3,2,4]=>11
[5,1,3,4,2]=>13
[5,1,4,2,3]=>13
[5,1,4,3,2]=>14
[5,2,1,3,4]=>11
[5,2,1,4,3]=>12
[5,2,3,1,4]=>13
[5,2,3,4,1]=>16
[5,2,4,1,3]=>15
[5,2,4,3,1]=>17
[5,3,1,2,4]=>13
[5,3,1,4,2]=>15
[5,3,2,1,4]=>14
[5,3,2,4,1]=>17
[5,3,4,1,2]=>18
[5,3,4,2,1]=>19
[5,4,1,2,3]=>16
[5,4,1,3,2]=>17
[5,4,2,1,3]=>17
[5,4,2,3,1]=>19
[5,4,3,1,2]=>19
[5,4,3,2,1]=>20
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>1
[1,2,3,5,4,6]=>1
[1,2,3,5,6,4]=>3
[1,2,3,6,4,5]=>3
[1,2,3,6,5,4]=>4
[1,2,4,3,5,6]=>1
[1,2,4,3,6,5]=>2
[1,2,4,5,3,6]=>3
[1,2,4,5,6,3]=>6
[1,2,4,6,3,5]=>5
[1,2,4,6,5,3]=>7
[1,2,5,3,4,6]=>3
[1,2,5,3,6,4]=>5
[1,2,5,4,3,6]=>4
[1,2,5,4,6,3]=>7
[1,2,5,6,3,4]=>8
[1,2,5,6,4,3]=>9
[1,2,6,3,4,5]=>6
[1,2,6,3,5,4]=>7
[1,2,6,4,3,5]=>7
[1,2,6,4,5,3]=>9
[1,2,6,5,3,4]=>9
[1,2,6,5,4,3]=>10
[1,3,2,4,5,6]=>1
[1,3,2,4,6,5]=>2
[1,3,2,5,4,6]=>2
[1,3,2,5,6,4]=>4
[1,3,2,6,4,5]=>4
[1,3,2,6,5,4]=>5
[1,3,4,2,5,6]=>3
[1,3,4,2,6,5]=>4
[1,3,4,5,2,6]=>6
[1,3,4,5,6,2]=>10
[1,3,4,6,2,5]=>8
[1,3,4,6,5,2]=>11
[1,3,5,2,4,6]=>5
[1,3,5,2,6,4]=>7
[1,3,5,4,2,6]=>7
[1,3,5,4,6,2]=>11
[1,3,5,6,2,4]=>11
[1,3,5,6,4,2]=>13
[1,3,6,2,4,5]=>8
[1,3,6,2,5,4]=>9
[1,3,6,4,2,5]=>10
[1,3,6,4,5,2]=>13
[1,3,6,5,2,4]=>12
[1,3,6,5,4,2]=>14
[1,4,2,3,5,6]=>3
[1,4,2,3,6,5]=>4
[1,4,2,5,3,6]=>5
[1,4,2,5,6,3]=>8
[1,4,2,6,3,5]=>7
[1,4,2,6,5,3]=>9
[1,4,3,2,5,6]=>4
[1,4,3,2,6,5]=>5
[1,4,3,5,2,6]=>7
[1,4,3,5,6,2]=>11
[1,4,3,6,2,5]=>9
[1,4,3,6,5,2]=>12
[1,4,5,2,3,6]=>8
[1,4,5,2,6,3]=>11
[1,4,5,3,2,6]=>9
[1,4,5,3,6,2]=>13
[1,4,5,6,2,3]=>15
[1,4,5,6,3,2]=>16
[1,4,6,2,3,5]=>11
[1,4,6,2,5,3]=>13
[1,4,6,3,2,5]=>12
[1,4,6,3,5,2]=>15
[1,4,6,5,2,3]=>16
[1,4,6,5,3,2]=>17
[1,5,2,3,4,6]=>6
[1,5,2,3,6,4]=>8
[1,5,2,4,3,6]=>7
[1,5,2,4,6,3]=>10
[1,5,2,6,3,4]=>11
[1,5,2,6,4,3]=>12
[1,5,3,2,4,6]=>7
[1,5,3,2,6,4]=>9
[1,5,3,4,2,6]=>9
[1,5,3,4,6,2]=>13
[1,5,3,6,2,4]=>13
[1,5,3,6,4,2]=>15
[1,5,4,2,3,6]=>9
[1,5,4,2,6,3]=>12
[1,5,4,3,2,6]=>10
[1,5,4,3,6,2]=>14
[1,5,4,6,2,3]=>16
[1,5,4,6,3,2]=>17
[1,5,6,2,3,4]=>15
[1,5,6,2,4,3]=>16
[1,5,6,3,2,4]=>16
[1,5,6,3,4,2]=>18
[1,5,6,4,2,3]=>18
[1,5,6,4,3,2]=>19
[1,6,2,3,4,5]=>10
[1,6,2,3,5,4]=>11
[1,6,2,4,3,5]=>11
[1,6,2,4,5,3]=>13
[1,6,2,5,3,4]=>13
[1,6,2,5,4,3]=>14
[1,6,3,2,4,5]=>11
[1,6,3,2,5,4]=>12
[1,6,3,4,2,5]=>13
[1,6,3,4,5,2]=>16
[1,6,3,5,2,4]=>15
[1,6,3,5,4,2]=>17
[1,6,4,2,3,5]=>13
[1,6,4,2,5,3]=>15
[1,6,4,3,2,5]=>14
[1,6,4,3,5,2]=>17
[1,6,4,5,2,3]=>18
[1,6,4,5,3,2]=>19
[1,6,5,2,3,4]=>16
[1,6,5,2,4,3]=>17
[1,6,5,3,2,4]=>17
[1,6,5,3,4,2]=>19
[1,6,5,4,2,3]=>19
[1,6,5,4,3,2]=>20
[2,1,3,4,5,6]=>1
[2,1,3,4,6,5]=>2
[2,1,3,5,4,6]=>2
[2,1,3,5,6,4]=>4
[2,1,3,6,4,5]=>4
[2,1,3,6,5,4]=>5
[2,1,4,3,5,6]=>2
[2,1,4,3,6,5]=>3
[2,1,4,5,3,6]=>4
[2,1,4,5,6,3]=>7
[2,1,4,6,3,5]=>6
[2,1,4,6,5,3]=>8
[2,1,5,3,4,6]=>4
[2,1,5,3,6,4]=>6
[2,1,5,4,3,6]=>5
[2,1,5,4,6,3]=>8
[2,1,5,6,3,4]=>9
[2,1,5,6,4,3]=>10
[2,1,6,3,4,5]=>7
[2,1,6,3,5,4]=>8
[2,1,6,4,3,5]=>8
[2,1,6,4,5,3]=>10
[2,1,6,5,3,4]=>10
[2,1,6,5,4,3]=>11
[2,3,1,4,5,6]=>3
[2,3,1,4,6,5]=>4
[2,3,1,5,4,6]=>4
[2,3,1,5,6,4]=>6
[2,3,1,6,4,5]=>6
[2,3,1,6,5,4]=>7
[2,3,4,1,5,6]=>6
[2,3,4,1,6,5]=>7
[2,3,4,5,1,6]=>10
[2,3,4,5,6,1]=>15
[2,3,4,6,1,5]=>12
[2,3,4,6,5,1]=>16
[2,3,5,1,4,6]=>8
[2,3,5,1,6,4]=>10
[2,3,5,4,1,6]=>11
[2,3,5,4,6,1]=>16
[2,3,5,6,1,4]=>15
[2,3,5,6,4,1]=>18
[2,3,6,1,4,5]=>11
[2,3,6,1,5,4]=>12
[2,3,6,4,1,5]=>14
[2,3,6,4,5,1]=>18
[2,3,6,5,1,4]=>16
[2,3,6,5,4,1]=>19
[2,4,1,3,5,6]=>5
[2,4,1,3,6,5]=>6
[2,4,1,5,3,6]=>7
[2,4,1,5,6,3]=>10
[2,4,1,6,3,5]=>9
[2,4,1,6,5,3]=>11
[2,4,3,1,5,6]=>7
[2,4,3,1,6,5]=>8
[2,4,3,5,1,6]=>11
[2,4,3,5,6,1]=>16
[2,4,3,6,1,5]=>13
[2,4,3,6,5,1]=>17
[2,4,5,1,3,6]=>11
[2,4,5,1,6,3]=>14
[2,4,5,3,1,6]=>13
[2,4,5,3,6,1]=>18
[2,4,5,6,1,3]=>19
[2,4,5,6,3,1]=>21
[2,4,6,1,3,5]=>14
[2,4,6,1,5,3]=>16
[2,4,6,3,1,5]=>16
[2,4,6,3,5,1]=>20
[2,4,6,5,1,3]=>20
[2,4,6,5,3,1]=>22
[2,5,1,3,4,6]=>8
[2,5,1,3,6,4]=>10
[2,5,1,4,3,6]=>9
[2,5,1,4,6,3]=>12
[2,5,1,6,3,4]=>13
[2,5,1,6,4,3]=>14
[2,5,3,1,4,6]=>10
[2,5,3,1,6,4]=>12
[2,5,3,4,1,6]=>13
[2,5,3,4,6,1]=>18
[2,5,3,6,1,4]=>17
[2,5,3,6,4,1]=>20
[2,5,4,1,3,6]=>12
[2,5,4,1,6,3]=>15
[2,5,4,3,1,6]=>14
[2,5,4,3,6,1]=>19
[2,5,4,6,1,3]=>20
[2,5,4,6,3,1]=>22
[2,5,6,1,3,4]=>18
[2,5,6,1,4,3]=>19
[2,5,6,3,1,4]=>20
[2,5,6,3,4,1]=>23
[2,5,6,4,1,3]=>22
[2,5,6,4,3,1]=>24
[2,6,1,3,4,5]=>12
[2,6,1,3,5,4]=>13
[2,6,1,4,3,5]=>13
[2,6,1,4,5,3]=>15
[2,6,1,5,3,4]=>15
[2,6,1,5,4,3]=>16
[2,6,3,1,4,5]=>14
[2,6,3,1,5,4]=>15
[2,6,3,4,1,5]=>17
[2,6,3,4,5,1]=>21
[2,6,3,5,1,4]=>19
[2,6,3,5,4,1]=>22
[2,6,4,1,3,5]=>16
[2,6,4,1,5,3]=>18
[2,6,4,3,1,5]=>18
[2,6,4,3,5,1]=>22
[2,6,4,5,1,3]=>22
[2,6,4,5,3,1]=>24
[2,6,5,1,3,4]=>19
[2,6,5,1,4,3]=>20
[2,6,5,3,1,4]=>21
[2,6,5,3,4,1]=>24
[2,6,5,4,1,3]=>23
[2,6,5,4,3,1]=>25
[3,1,2,4,5,6]=>3
[3,1,2,4,6,5]=>4
[3,1,2,5,4,6]=>4
[3,1,2,5,6,4]=>6
[3,1,2,6,4,5]=>6
[3,1,2,6,5,4]=>7
[3,1,4,2,5,6]=>5
[3,1,4,2,6,5]=>6
[3,1,4,5,2,6]=>8
[3,1,4,5,6,2]=>12
[3,1,4,6,2,5]=>10
[3,1,4,6,5,2]=>13
[3,1,5,2,4,6]=>7
[3,1,5,2,6,4]=>9
[3,1,5,4,2,6]=>9
[3,1,5,4,6,2]=>13
[3,1,5,6,2,4]=>13
[3,1,5,6,4,2]=>15
[3,1,6,2,4,5]=>10
[3,1,6,2,5,4]=>11
[3,1,6,4,2,5]=>12
[3,1,6,4,5,2]=>15
[3,1,6,5,2,4]=>14
[3,1,6,5,4,2]=>16
[3,2,1,4,5,6]=>4
[3,2,1,4,6,5]=>5
[3,2,1,5,4,6]=>5
[3,2,1,5,6,4]=>7
[3,2,1,6,4,5]=>7
[3,2,1,6,5,4]=>8
[3,2,4,1,5,6]=>7
[3,2,4,1,6,5]=>8
[3,2,4,5,1,6]=>11
[3,2,4,5,6,1]=>16
[3,2,4,6,1,5]=>13
[3,2,4,6,5,1]=>17
[3,2,5,1,4,6]=>9
[3,2,5,1,6,4]=>11
[3,2,5,4,1,6]=>12
[3,2,5,4,6,1]=>17
[3,2,5,6,1,4]=>16
[3,2,5,6,4,1]=>19
[3,2,6,1,4,5]=>12
[3,2,6,1,5,4]=>13
[3,2,6,4,1,5]=>15
[3,2,6,4,5,1]=>19
[3,2,6,5,1,4]=>17
[3,2,6,5,4,1]=>20
[3,4,1,2,5,6]=>8
[3,4,1,2,6,5]=>9
[3,4,1,5,2,6]=>11
[3,4,1,5,6,2]=>15
[3,4,1,6,2,5]=>13
[3,4,1,6,5,2]=>16
[3,4,2,1,5,6]=>9
[3,4,2,1,6,5]=>10
[3,4,2,5,1,6]=>13
[3,4,2,5,6,1]=>18
[3,4,2,6,1,5]=>15
[3,4,2,6,5,1]=>19
[3,4,5,1,2,6]=>15
[3,4,5,1,6,2]=>19
[3,4,5,2,1,6]=>16
[3,4,5,2,6,1]=>21
[3,4,5,6,1,2]=>24
[3,4,5,6,2,1]=>25
[3,4,6,1,2,5]=>18
[3,4,6,1,5,2]=>21
[3,4,6,2,1,5]=>19
[3,4,6,2,5,1]=>23
[3,4,6,5,1,2]=>25
[3,4,6,5,2,1]=>26
[3,5,1,2,4,6]=>11
[3,5,1,2,6,4]=>13
[3,5,1,4,2,6]=>13
[3,5,1,4,6,2]=>17
[3,5,1,6,2,4]=>17
[3,5,1,6,4,2]=>19
[3,5,2,1,4,6]=>12
[3,5,2,1,6,4]=>14
[3,5,2,4,1,6]=>15
[3,5,2,4,6,1]=>20
[3,5,2,6,1,4]=>19
[3,5,2,6,4,1]=>22
[3,5,4,1,2,6]=>16
[3,5,4,1,6,2]=>20
[3,5,4,2,1,6]=>17
[3,5,4,2,6,1]=>22
[3,5,4,6,1,2]=>25
[3,5,4,6,2,1]=>26
[3,5,6,1,2,4]=>22
[3,5,6,1,4,2]=>24
[3,5,6,2,1,4]=>23
[3,5,6,2,4,1]=>26
[3,5,6,4,1,2]=>27
[3,5,6,4,2,1]=>28
[3,6,1,2,4,5]=>15
[3,6,1,2,5,4]=>16
[3,6,1,4,2,5]=>17
[3,6,1,4,5,2]=>20
[3,6,1,5,2,4]=>19
[3,6,1,5,4,2]=>21
[3,6,2,1,4,5]=>16
[3,6,2,1,5,4]=>17
[3,6,2,4,1,5]=>19
[3,6,2,4,5,1]=>23
[3,6,2,5,1,4]=>21
[3,6,2,5,4,1]=>24
[3,6,4,1,2,5]=>20
[3,6,4,1,5,2]=>23
[3,6,4,2,1,5]=>21
[3,6,4,2,5,1]=>25
[3,6,4,5,1,2]=>27
[3,6,4,5,2,1]=>28
[3,6,5,1,2,4]=>23
[3,6,5,1,4,2]=>25
[3,6,5,2,1,4]=>24
[3,6,5,2,4,1]=>27
[3,6,5,4,1,2]=>28
[3,6,5,4,2,1]=>29
[4,1,2,3,5,6]=>6
[4,1,2,3,6,5]=>7
[4,1,2,5,3,6]=>8
[4,1,2,5,6,3]=>11
[4,1,2,6,3,5]=>10
[4,1,2,6,5,3]=>12
[4,1,3,2,5,6]=>7
[4,1,3,2,6,5]=>8
[4,1,3,5,2,6]=>10
[4,1,3,5,6,2]=>14
[4,1,3,6,2,5]=>12
[4,1,3,6,5,2]=>15
[4,1,5,2,3,6]=>11
[4,1,5,2,6,3]=>14
[4,1,5,3,2,6]=>12
[4,1,5,3,6,2]=>16
[4,1,5,6,2,3]=>18
[4,1,5,6,3,2]=>19
[4,1,6,2,3,5]=>14
[4,1,6,2,5,3]=>16
[4,1,6,3,2,5]=>15
[4,1,6,3,5,2]=>18
[4,1,6,5,2,3]=>19
[4,1,6,5,3,2]=>20
[4,2,1,3,5,6]=>7
[4,2,1,3,6,5]=>8
[4,2,1,5,3,6]=>9
[4,2,1,5,6,3]=>12
[4,2,1,6,3,5]=>11
[4,2,1,6,5,3]=>13
[4,2,3,1,5,6]=>9
[4,2,3,1,6,5]=>10
[4,2,3,5,1,6]=>13
[4,2,3,5,6,1]=>18
[4,2,3,6,1,5]=>15
[4,2,3,6,5,1]=>19
[4,2,5,1,3,6]=>13
[4,2,5,1,6,3]=>16
[4,2,5,3,1,6]=>15
[4,2,5,3,6,1]=>20
[4,2,5,6,1,3]=>21
[4,2,5,6,3,1]=>23
[4,2,6,1,3,5]=>16
[4,2,6,1,5,3]=>18
[4,2,6,3,1,5]=>18
[4,2,6,3,5,1]=>22
[4,2,6,5,1,3]=>22
[4,2,6,5,3,1]=>24
[4,3,1,2,5,6]=>9
[4,3,1,2,6,5]=>10
[4,3,1,5,2,6]=>12
[4,3,1,5,6,2]=>16
[4,3,1,6,2,5]=>14
[4,3,1,6,5,2]=>17
[4,3,2,1,5,6]=>10
[4,3,2,1,6,5]=>11
[4,3,2,5,1,6]=>14
[4,3,2,5,6,1]=>19
[4,3,2,6,1,5]=>16
[4,3,2,6,5,1]=>20
[4,3,5,1,2,6]=>16
[4,3,5,1,6,2]=>20
[4,3,5,2,1,6]=>17
[4,3,5,2,6,1]=>22
[4,3,5,6,1,2]=>25
[4,3,5,6,2,1]=>26
[4,3,6,1,2,5]=>19
[4,3,6,1,5,2]=>22
[4,3,6,2,1,5]=>20
[4,3,6,2,5,1]=>24
[4,3,6,5,1,2]=>26
[4,3,6,5,2,1]=>27
[4,5,1,2,3,6]=>15
[4,5,1,2,6,3]=>18
[4,5,1,3,2,6]=>16
[4,5,1,3,6,2]=>20
[4,5,1,6,2,3]=>22
[4,5,1,6,3,2]=>23
[4,5,2,1,3,6]=>16
[4,5,2,1,6,3]=>19
[4,5,2,3,1,6]=>18
[4,5,2,3,6,1]=>23
[4,5,2,6,1,3]=>24
[4,5,2,6,3,1]=>26
[4,5,3,1,2,6]=>18
[4,5,3,1,6,2]=>22
[4,5,3,2,1,6]=>19
[4,5,3,2,6,1]=>24
[4,5,3,6,1,2]=>27
[4,5,3,6,2,1]=>28
[4,5,6,1,2,3]=>27
[4,5,6,1,3,2]=>28
[4,5,6,2,1,3]=>28
[4,5,6,2,3,1]=>30
[4,5,6,3,1,2]=>30
[4,5,6,3,2,1]=>31
[4,6,1,2,3,5]=>19
[4,6,1,2,5,3]=>21
[4,6,1,3,2,5]=>20
[4,6,1,3,5,2]=>23
[4,6,1,5,2,3]=>24
[4,6,1,5,3,2]=>25
[4,6,2,1,3,5]=>20
[4,6,2,1,5,3]=>22
[4,6,2,3,1,5]=>22
[4,6,2,3,5,1]=>26
[4,6,2,5,1,3]=>26
[4,6,2,5,3,1]=>28
[4,6,3,1,2,5]=>22
[4,6,3,1,5,2]=>25
[4,6,3,2,1,5]=>23
[4,6,3,2,5,1]=>27
[4,6,3,5,1,2]=>29
[4,6,3,5,2,1]=>30
[4,6,5,1,2,3]=>28
[4,6,5,1,3,2]=>29
[4,6,5,2,1,3]=>29
[4,6,5,2,3,1]=>31
[4,6,5,3,1,2]=>31
[4,6,5,3,2,1]=>32
[5,1,2,3,4,6]=>10
[5,1,2,3,6,4]=>12
[5,1,2,4,3,6]=>11
[5,1,2,4,6,3]=>14
[5,1,2,6,3,4]=>15
[5,1,2,6,4,3]=>16
[5,1,3,2,4,6]=>11
[5,1,3,2,6,4]=>13
[5,1,3,4,2,6]=>13
[5,1,3,4,6,2]=>17
[5,1,3,6,2,4]=>17
[5,1,3,6,4,2]=>19
[5,1,4,2,3,6]=>13
[5,1,4,2,6,3]=>16
[5,1,4,3,2,6]=>14
[5,1,4,3,6,2]=>18
[5,1,4,6,2,3]=>20
[5,1,4,6,3,2]=>21
[5,1,6,2,3,4]=>19
[5,1,6,2,4,3]=>20
[5,1,6,3,2,4]=>20
[5,1,6,3,4,2]=>22
[5,1,6,4,2,3]=>22
[5,1,6,4,3,2]=>23
[5,2,1,3,4,6]=>11
[5,2,1,3,6,4]=>13
[5,2,1,4,3,6]=>12
[5,2,1,4,6,3]=>15
[5,2,1,6,3,4]=>16
[5,2,1,6,4,3]=>17
[5,2,3,1,4,6]=>13
[5,2,3,1,6,4]=>15
[5,2,3,4,1,6]=>16
[5,2,3,4,6,1]=>21
[5,2,3,6,1,4]=>20
[5,2,3,6,4,1]=>23
[5,2,4,1,3,6]=>15
[5,2,4,1,6,3]=>18
[5,2,4,3,1,6]=>17
[5,2,4,3,6,1]=>22
[5,2,4,6,1,3]=>23
[5,2,4,6,3,1]=>25
[5,2,6,1,3,4]=>21
[5,2,6,1,4,3]=>22
[5,2,6,3,1,4]=>23
[5,2,6,3,4,1]=>26
[5,2,6,4,1,3]=>25
[5,2,6,4,3,1]=>27
[5,3,1,2,4,6]=>13
[5,3,1,2,6,4]=>15
[5,3,1,4,2,6]=>15
[5,3,1,4,6,2]=>19
[5,3,1,6,2,4]=>19
[5,3,1,6,4,2]=>21
[5,3,2,1,4,6]=>14
[5,3,2,1,6,4]=>16
[5,3,2,4,1,6]=>17
[5,3,2,4,6,1]=>22
[5,3,2,6,1,4]=>21
[5,3,2,6,4,1]=>24
[5,3,4,1,2,6]=>18
[5,3,4,1,6,2]=>22
[5,3,4,2,1,6]=>19
[5,3,4,2,6,1]=>24
[5,3,4,6,1,2]=>27
[5,3,4,6,2,1]=>28
[5,3,6,1,2,4]=>24
[5,3,6,1,4,2]=>26
[5,3,6,2,1,4]=>25
[5,3,6,2,4,1]=>28
[5,3,6,4,1,2]=>29
[5,3,6,4,2,1]=>30
[5,4,1,2,3,6]=>16
[5,4,1,2,6,3]=>19
[5,4,1,3,2,6]=>17
[5,4,1,3,6,2]=>21
[5,4,1,6,2,3]=>23
[5,4,1,6,3,2]=>24
[5,4,2,1,3,6]=>17
[5,4,2,1,6,3]=>20
[5,4,2,3,1,6]=>19
[5,4,2,3,6,1]=>24
[5,4,2,6,1,3]=>25
[5,4,2,6,3,1]=>27
[5,4,3,1,2,6]=>19
[5,4,3,1,6,2]=>23
[5,4,3,2,1,6]=>20
[5,4,3,2,6,1]=>25
[5,4,3,6,1,2]=>28
[5,4,3,6,2,1]=>29
[5,4,6,1,2,3]=>28
[5,4,6,1,3,2]=>29
[5,4,6,2,1,3]=>29
[5,4,6,2,3,1]=>31
[5,4,6,3,1,2]=>31
[5,4,6,3,2,1]=>32
[5,6,1,2,3,4]=>24
[5,6,1,2,4,3]=>25
[5,6,1,3,2,4]=>25
[5,6,1,3,4,2]=>27
[5,6,1,4,2,3]=>27
[5,6,1,4,3,2]=>28
[5,6,2,1,3,4]=>25
[5,6,2,1,4,3]=>26
[5,6,2,3,1,4]=>27
[5,6,2,3,4,1]=>30
[5,6,2,4,1,3]=>29
[5,6,2,4,3,1]=>31
[5,6,3,1,2,4]=>27
[5,6,3,1,4,2]=>29
[5,6,3,2,1,4]=>28
[5,6,3,2,4,1]=>31
[5,6,3,4,1,2]=>32
[5,6,3,4,2,1]=>33
[5,6,4,1,2,3]=>30
[5,6,4,1,3,2]=>31
[5,6,4,2,1,3]=>31
[5,6,4,2,3,1]=>33
[5,6,4,3,1,2]=>33
[5,6,4,3,2,1]=>34
[6,1,2,3,4,5]=>15
[6,1,2,3,5,4]=>16
[6,1,2,4,3,5]=>16
[6,1,2,4,5,3]=>18
[6,1,2,5,3,4]=>18
[6,1,2,5,4,3]=>19
[6,1,3,2,4,5]=>16
[6,1,3,2,5,4]=>17
[6,1,3,4,2,5]=>18
[6,1,3,4,5,2]=>21
[6,1,3,5,2,4]=>20
[6,1,3,5,4,2]=>22
[6,1,4,2,3,5]=>18
[6,1,4,2,5,3]=>20
[6,1,4,3,2,5]=>19
[6,1,4,3,5,2]=>22
[6,1,4,5,2,3]=>23
[6,1,4,5,3,2]=>24
[6,1,5,2,3,4]=>21
[6,1,5,2,4,3]=>22
[6,1,5,3,2,4]=>22
[6,1,5,3,4,2]=>24
[6,1,5,4,2,3]=>24
[6,1,5,4,3,2]=>25
[6,2,1,3,4,5]=>16
[6,2,1,3,5,4]=>17
[6,2,1,4,3,5]=>17
[6,2,1,4,5,3]=>19
[6,2,1,5,3,4]=>19
[6,2,1,5,4,3]=>20
[6,2,3,1,4,5]=>18
[6,2,3,1,5,4]=>19
[6,2,3,4,1,5]=>21
[6,2,3,4,5,1]=>25
[6,2,3,5,1,4]=>23
[6,2,3,5,4,1]=>26
[6,2,4,1,3,5]=>20
[6,2,4,1,5,3]=>22
[6,2,4,3,1,5]=>22
[6,2,4,3,5,1]=>26
[6,2,4,5,1,3]=>26
[6,2,4,5,3,1]=>28
[6,2,5,1,3,4]=>23
[6,2,5,1,4,3]=>24
[6,2,5,3,1,4]=>25
[6,2,5,3,4,1]=>28
[6,2,5,4,1,3]=>27
[6,2,5,4,3,1]=>29
[6,3,1,2,4,5]=>18
[6,3,1,2,5,4]=>19
[6,3,1,4,2,5]=>20
[6,3,1,4,5,2]=>23
[6,3,1,5,2,4]=>22
[6,3,1,5,4,2]=>24
[6,3,2,1,4,5]=>19
[6,3,2,1,5,4]=>20
[6,3,2,4,1,5]=>22
[6,3,2,4,5,1]=>26
[6,3,2,5,1,4]=>24
[6,3,2,5,4,1]=>27
[6,3,4,1,2,5]=>23
[6,3,4,1,5,2]=>26
[6,3,4,2,1,5]=>24
[6,3,4,2,5,1]=>28
[6,3,4,5,1,2]=>30
[6,3,4,5,2,1]=>31
[6,3,5,1,2,4]=>26
[6,3,5,1,4,2]=>28
[6,3,5,2,1,4]=>27
[6,3,5,2,4,1]=>30
[6,3,5,4,1,2]=>31
[6,3,5,4,2,1]=>32
[6,4,1,2,3,5]=>21
[6,4,1,2,5,3]=>23
[6,4,1,3,2,5]=>22
[6,4,1,3,5,2]=>25
[6,4,1,5,2,3]=>26
[6,4,1,5,3,2]=>27
[6,4,2,1,3,5]=>22
[6,4,2,1,5,3]=>24
[6,4,2,3,1,5]=>24
[6,4,2,3,5,1]=>28
[6,4,2,5,1,3]=>28
[6,4,2,5,3,1]=>30
[6,4,3,1,2,5]=>24
[6,4,3,1,5,2]=>27
[6,4,3,2,1,5]=>25
[6,4,3,2,5,1]=>29
[6,4,3,5,1,2]=>31
[6,4,3,5,2,1]=>32
[6,4,5,1,2,3]=>30
[6,4,5,1,3,2]=>31
[6,4,5,2,1,3]=>31
[6,4,5,2,3,1]=>33
[6,4,5,3,1,2]=>33
[6,4,5,3,2,1]=>34
[6,5,1,2,3,4]=>25
[6,5,1,2,4,3]=>26
[6,5,1,3,2,4]=>26
[6,5,1,3,4,2]=>28
[6,5,1,4,2,3]=>28
[6,5,1,4,3,2]=>29
[6,5,2,1,3,4]=>26
[6,5,2,1,4,3]=>27
[6,5,2,3,1,4]=>28
[6,5,2,3,4,1]=>31
[6,5,2,4,1,3]=>30
[6,5,2,4,3,1]=>32
[6,5,3,1,2,4]=>28
[6,5,3,1,4,2]=>30
[6,5,3,2,1,4]=>29
[6,5,3,2,4,1]=>32
[6,5,3,4,1,2]=>33
[6,5,3,4,2,1]=>34
[6,5,4,1,2,3]=>31
[6,5,4,1,3,2]=>32
[6,5,4,2,1,3]=>32
[6,5,4,2,3,1]=>34
[6,5,4,3,1,2]=>34
[6,5,4,3,2,1]=>35
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Description
The inversion sum of a permutation.
A pair $a < b$ is an inversion of a permutation $\pi$ if $\pi(a) > \pi(b)$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
This is also half of the metric associated with Spearmans coefficient of association $\rho$, $\sum_i (\pi_i - i)^2$, see [5].
This is also equal to the total number of occurrences of the classical permutation patterns $[2,1], [2, 3, 1], [3, 1, 2]$, and $[3, 2, 1]$, see [2].
This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3].
This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding monotone triangle and subtracting $\binom{n+2}{3}$, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of $n$.
This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation $\pi$ in Bruhat order, see Theorem 1 of [6].
A pair $a < b$ is an inversion of a permutation $\pi$ if $\pi(a) > \pi(b)$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
This is also half of the metric associated with Spearmans coefficient of association $\rho$, $\sum_i (\pi_i - i)^2$, see [5].
This is also equal to the total number of occurrences of the classical permutation patterns $[2,1], [2, 3, 1], [3, 1, 2]$, and $[3, 2, 1]$, see [2].
This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3].
This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding monotone triangle and subtracting $\binom{n+2}{3}$, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of $n$.
This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation $\pi$ in Bruhat order, see Theorem 1 of [6].
References
[1] Lascoux, A., Schützenberger, M.-P. Treillis et bases des groupes de Coxeter MathSciNet:1395667
[2] Sack, J., Úlfarsson, H. Refined inversion statistics on permutations MathSciNet:2880660 arXiv:1106.1995
[3] Striker, J. A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux MathSciNet:2794039
[4] a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers. OEIS:A135298
[5] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray MathSciNet:0652736
[6] Kobayashi, M. Enumeration of bigrassmannian permutations below a permutation in Bruhat order arXiv:1005.3335
[2] Sack, J., Úlfarsson, H. Refined inversion statistics on permutations MathSciNet:2880660 arXiv:1106.1995
[3] Striker, J. A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux MathSciNet:2794039
[4] a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers. OEIS:A135298
[5] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray MathSciNet:0652736
[6] Kobayashi, M. Enumeration of bigrassmannian permutations below a permutation in Bruhat order arXiv:1005.3335
Code
def statistic(pi):
return sum( inv[1]-inv[0] for inv in pi.inversions() )
def statistic_alternative(perm):
pmatrix = perm.to_matrix()
w = AlternatingSignMatrix(pmatrix).to_monotone_triangle()
counter = -binomial(len(perm)+2,3)
for k in [0..(len(w)-1)]:
for j in [0..(len(w[k])-1)]:
counter=counter+w[k][j]
return counter
Created
Mar 25, 2013 at 20:49 by Jessica Striker
Updated
May 30, 2019 at 11:28 by Masato Kobayashi
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