Identifier
- St000500: Permutations ⟶ ℤ
Values
=>
[]=>0
[1]=>1
[1,2]=>4
[2,1]=>0
[1,2,3]=>9
[1,3,2]=>4
[2,1,3]=>0
[2,3,1]=>4
[3,1,2]=>0
[3,2,1]=>1
[1,2,3,4]=>16
[1,2,4,3]=>10
[1,3,2,4]=>6
[1,3,4,2]=>10
[1,4,2,3]=>6
[1,4,3,2]=>6
[2,1,3,4]=>0
[2,1,4,3]=>0
[2,3,1,4]=>6
[2,3,4,1]=>10
[2,4,1,3]=>4
[2,4,3,1]=>6
[3,1,2,4]=>0
[3,1,4,2]=>0
[3,2,1,4]=>2
[3,2,4,1]=>0
[3,4,1,2]=>4
[3,4,2,1]=>6
[4,1,2,3]=>0
[4,1,3,2]=>0
[4,2,1,3]=>2
[4,2,3,1]=>0
[4,3,1,2]=>2
[4,3,2,1]=>0
[1,2,3,4,5]=>25
[1,2,3,5,4]=>18
[1,2,4,3,5]=>14
[1,2,4,5,3]=>18
[1,2,5,3,4]=>14
[1,2,5,4,3]=>13
[1,3,2,4,5]=>8
[1,3,2,5,4]=>7
[1,3,4,2,5]=>14
[1,3,4,5,2]=>18
[1,3,5,2,4]=>11
[1,3,5,4,2]=>13
[1,4,2,3,5]=>8
[1,4,2,5,3]=>7
[1,4,3,2,5]=>9
[1,4,3,5,2]=>7
[1,4,5,2,3]=>11
[1,4,5,3,2]=>13
[1,5,2,3,4]=>8
[1,5,2,4,3]=>7
[1,5,3,2,4]=>9
[1,5,3,4,2]=>7
[1,5,4,2,3]=>9
[1,5,4,3,2]=>6
[2,1,3,4,5]=>0
[2,1,3,5,4]=>0
[2,1,4,3,5]=>0
[2,1,4,5,3]=>0
[2,1,5,3,4]=>0
[2,1,5,4,3]=>0
[2,3,1,4,5]=>8
[2,3,1,5,4]=>7
[2,3,4,1,5]=>14
[2,3,4,5,1]=>18
[2,3,5,1,4]=>11
[2,3,5,4,1]=>13
[2,4,1,3,5]=>5
[2,4,1,5,3]=>7
[2,4,3,1,5]=>9
[2,4,3,5,1]=>7
[2,4,5,1,3]=>11
[2,4,5,3,1]=>13
[2,5,1,3,4]=>5
[2,5,1,4,3]=>5
[2,5,3,1,4]=>9
[2,5,3,4,1]=>7
[2,5,4,1,3]=>7
[2,5,4,3,1]=>6
[3,1,2,4,5]=>0
[3,1,2,5,4]=>0
[3,1,4,2,5]=>0
[3,1,4,5,2]=>0
[3,1,5,2,4]=>0
[3,1,5,4,2]=>0
[3,2,1,4,5]=>3
[3,2,1,5,4]=>3
[3,2,4,1,5]=>0
[3,2,4,5,1]=>0
[3,2,5,1,4]=>0
[3,2,5,4,1]=>0
[3,4,1,2,5]=>5
[3,4,1,5,2]=>7
[3,4,2,1,5]=>9
[3,4,2,5,1]=>7
[3,4,5,1,2]=>11
[3,4,5,2,1]=>13
[3,5,1,2,4]=>5
[3,5,1,4,2]=>5
[3,5,2,1,4]=>7
[3,5,2,4,1]=>5
[3,5,4,1,2]=>7
[3,5,4,2,1]=>6
[4,1,2,3,5]=>0
[4,1,2,5,3]=>0
[4,1,3,2,5]=>0
[4,1,3,5,2]=>0
[4,1,5,2,3]=>0
[4,1,5,3,2]=>0
[4,2,1,3,5]=>3
[4,2,1,5,3]=>3
[4,2,3,1,5]=>0
[4,2,3,5,1]=>0
[4,2,5,1,3]=>0
[4,2,5,3,1]=>0
[4,3,1,2,5]=>3
[4,3,1,5,2]=>3
[4,3,2,1,5]=>0
[4,3,2,5,1]=>2
[4,3,5,1,2]=>0
[4,3,5,2,1]=>0
[4,5,1,2,3]=>5
[4,5,1,3,2]=>5
[4,5,2,1,3]=>7
[4,5,2,3,1]=>5
[4,5,3,1,2]=>7
[4,5,3,2,1]=>6
[5,1,2,3,4]=>0
[5,1,2,4,3]=>0
[5,1,3,2,4]=>0
[5,1,3,4,2]=>0
[5,1,4,2,3]=>0
[5,1,4,3,2]=>0
[5,2,1,3,4]=>3
[5,2,1,4,3]=>3
[5,2,3,1,4]=>0
[5,2,3,4,1]=>0
[5,2,4,1,3]=>0
[5,2,4,3,1]=>0
[5,3,1,2,4]=>3
[5,3,1,4,2]=>3
[5,3,2,1,4]=>0
[5,3,2,4,1]=>2
[5,3,4,1,2]=>0
[5,3,4,2,1]=>0
[5,4,1,2,3]=>3
[5,4,1,3,2]=>2
[5,4,2,1,3]=>0
[5,4,2,3,1]=>2
[5,4,3,1,2]=>0
[5,4,3,2,1]=>1
[1,2,3,4,5,6]=>36
[1,2,3,4,6,5]=>28
[1,2,3,5,4,6]=>24
[1,2,3,5,6,4]=>28
[1,2,3,6,4,5]=>24
[1,2,3,6,5,4]=>22
[1,2,4,3,5,6]=>18
[1,2,4,3,6,5]=>16
[1,2,4,5,3,6]=>24
[1,2,4,5,6,3]=>28
[1,2,4,6,3,5]=>20
[1,2,4,6,5,3]=>22
[1,2,5,3,4,6]=>18
[1,2,5,3,6,4]=>16
[1,2,5,4,3,6]=>18
[1,2,5,4,6,3]=>16
[1,2,5,6,3,4]=>20
[1,2,5,6,4,3]=>22
[1,2,6,3,4,5]=>18
[1,2,6,3,5,4]=>16
[1,2,6,4,3,5]=>18
[1,2,6,4,5,3]=>16
[1,2,6,5,3,4]=>18
[1,2,6,5,4,3]=>14
[1,3,2,4,5,6]=>10
[1,3,2,4,6,5]=>9
[1,3,2,5,4,6]=>9
[1,3,2,5,6,4]=>9
[1,3,2,6,4,5]=>9
[1,3,2,6,5,4]=>8
[1,3,4,2,5,6]=>18
[1,3,4,2,6,5]=>16
[1,3,4,5,2,6]=>24
[1,3,4,5,6,2]=>28
[1,3,4,6,2,5]=>20
[1,3,4,6,5,2]=>22
[1,3,5,2,4,6]=>14
[1,3,5,2,6,4]=>16
[1,3,5,4,2,6]=>18
[1,3,5,4,6,2]=>16
[1,3,5,6,2,4]=>20
[1,3,5,6,4,2]=>22
[1,3,6,2,4,5]=>14
[1,3,6,2,5,4]=>13
[1,3,6,4,2,5]=>18
[1,3,6,4,5,2]=>16
[1,3,6,5,2,4]=>15
[1,3,6,5,4,2]=>14
[1,4,2,3,5,6]=>10
[1,4,2,3,6,5]=>9
[1,4,2,5,3,6]=>9
[1,4,2,5,6,3]=>9
[1,4,2,6,3,5]=>9
[1,4,2,6,5,3]=>8
[1,4,3,2,5,6]=>12
[1,4,3,2,6,5]=>11
[1,4,3,5,2,6]=>9
[1,4,3,5,6,2]=>9
[1,4,3,6,2,5]=>8
[1,4,3,6,5,2]=>8
[1,4,5,2,3,6]=>14
[1,4,5,2,6,3]=>16
[1,4,5,3,2,6]=>18
[1,4,5,3,6,2]=>16
[1,4,5,6,2,3]=>20
[1,4,5,6,3,2]=>22
[1,4,6,2,3,5]=>14
[1,4,6,2,5,3]=>13
[1,4,6,3,2,5]=>15
[1,4,6,3,5,2]=>13
[1,4,6,5,2,3]=>15
[1,4,6,5,3,2]=>14
[1,5,2,3,4,6]=>10
[1,5,2,3,6,4]=>9
[1,5,2,4,3,6]=>9
[1,5,2,4,6,3]=>9
[1,5,2,6,3,4]=>9
[1,5,2,6,4,3]=>8
[1,5,3,2,4,6]=>12
[1,5,3,2,6,4]=>11
[1,5,3,4,2,6]=>9
[1,5,3,4,6,2]=>9
[1,5,3,6,2,4]=>8
[1,5,3,6,4,2]=>8
[1,5,4,2,3,6]=>12
[1,5,4,2,6,3]=>11
[1,5,4,3,2,6]=>8
[1,5,4,3,6,2]=>10
[1,5,4,6,2,3]=>8
[1,5,4,6,3,2]=>8
[1,5,6,2,3,4]=>14
[1,5,6,2,4,3]=>13
[1,5,6,3,2,4]=>15
[1,5,6,3,4,2]=>13
[1,5,6,4,2,3]=>15
[1,5,6,4,3,2]=>14
[1,6,2,3,4,5]=>10
[1,6,2,3,5,4]=>9
[1,6,2,4,3,5]=>9
[1,6,2,4,5,3]=>9
[1,6,2,5,3,4]=>9
[1,6,2,5,4,3]=>8
[1,6,3,2,4,5]=>12
[1,6,3,2,5,4]=>11
[1,6,3,4,2,5]=>9
[1,6,3,4,5,2]=>9
[1,6,3,5,2,4]=>8
[1,6,3,5,4,2]=>8
[1,6,4,2,3,5]=>12
[1,6,4,2,5,3]=>11
[1,6,4,3,2,5]=>8
[1,6,4,3,5,2]=>10
[1,6,4,5,2,3]=>8
[1,6,4,5,3,2]=>8
[1,6,5,2,3,4]=>12
[1,6,5,2,4,3]=>10
[1,6,5,3,2,4]=>8
[1,6,5,3,4,2]=>10
[1,6,5,4,2,3]=>8
[1,6,5,4,3,2]=>8
[2,1,3,4,5,6]=>0
[2,1,3,4,6,5]=>0
[2,1,3,5,4,6]=>0
[2,1,3,5,6,4]=>0
[2,1,3,6,4,5]=>0
[2,1,3,6,5,4]=>0
[2,1,4,3,5,6]=>0
[2,1,4,3,6,5]=>0
[2,1,4,5,3,6]=>0
[2,1,4,5,6,3]=>0
[2,1,4,6,3,5]=>0
[2,1,4,6,5,3]=>0
[2,1,5,3,4,6]=>0
[2,1,5,3,6,4]=>0
[2,1,5,4,3,6]=>0
[2,1,5,4,6,3]=>0
[2,1,5,6,3,4]=>0
[2,1,5,6,4,3]=>0
[2,1,6,3,4,5]=>0
[2,1,6,3,5,4]=>0
[2,1,6,4,3,5]=>0
[2,1,6,4,5,3]=>0
[2,1,6,5,3,4]=>0
[2,1,6,5,4,3]=>0
[2,3,1,4,5,6]=>10
[2,3,1,4,6,5]=>9
[2,3,1,5,4,6]=>9
[2,3,1,5,6,4]=>9
[2,3,1,6,4,5]=>9
[2,3,1,6,5,4]=>8
[2,3,4,1,5,6]=>18
[2,3,4,1,6,5]=>16
[2,3,4,5,1,6]=>24
[2,3,4,5,6,1]=>28
[2,3,4,6,1,5]=>20
[2,3,4,6,5,1]=>22
[2,3,5,1,4,6]=>14
[2,3,5,1,6,4]=>16
[2,3,5,4,1,6]=>18
[2,3,5,4,6,1]=>16
[2,3,5,6,1,4]=>20
[2,3,5,6,4,1]=>22
[2,3,6,1,4,5]=>14
[2,3,6,1,5,4]=>13
[2,3,6,4,1,5]=>18
[2,3,6,4,5,1]=>16
[2,3,6,5,1,4]=>15
[2,3,6,5,4,1]=>14
[2,4,1,3,5,6]=>6
[2,4,1,3,6,5]=>7
[2,4,1,5,3,6]=>9
[2,4,1,5,6,3]=>9
[2,4,1,6,3,5]=>7
[2,4,1,6,5,3]=>8
[2,4,3,1,5,6]=>12
[2,4,3,1,6,5]=>11
[2,4,3,5,1,6]=>9
[2,4,3,5,6,1]=>9
[2,4,3,6,1,5]=>8
[2,4,3,6,5,1]=>8
[2,4,5,1,3,6]=>14
[2,4,5,1,6,3]=>16
[2,4,5,3,1,6]=>18
[2,4,5,3,6,1]=>16
[2,4,5,6,1,3]=>20
[2,4,5,6,3,1]=>22
[2,4,6,1,3,5]=>12
[2,4,6,1,5,3]=>13
[2,4,6,3,1,5]=>15
[2,4,6,3,5,1]=>13
[2,4,6,5,1,3]=>15
[2,4,6,5,3,1]=>14
[2,5,1,3,4,6]=>6
[2,5,1,3,6,4]=>7
[2,5,1,4,3,6]=>6
[2,5,1,4,6,3]=>6
[2,5,1,6,3,4]=>7
[2,5,1,6,4,3]=>8
[2,5,3,1,4,6]=>12
[2,5,3,1,6,4]=>11
[2,5,3,4,1,6]=>9
[2,5,3,4,6,1]=>9
[2,5,3,6,1,4]=>8
[2,5,3,6,4,1]=>8
[2,5,4,1,3,6]=>9
[2,5,4,1,6,3]=>11
[2,5,4,3,1,6]=>8
[2,5,4,3,6,1]=>10
[2,5,4,6,1,3]=>8
[2,5,4,6,3,1]=>8
[2,5,6,1,3,4]=>12
[2,5,6,1,4,3]=>13
[2,5,6,3,1,4]=>15
[2,5,6,3,4,1]=>13
[2,5,6,4,1,3]=>15
[2,5,6,4,3,1]=>14
[2,6,1,3,4,5]=>6
[2,6,1,3,5,4]=>6
[2,6,1,4,3,5]=>6
[2,6,1,4,5,3]=>6
[2,6,1,5,3,4]=>6
[2,6,1,5,4,3]=>6
[2,6,3,1,4,5]=>12
[2,6,3,1,5,4]=>11
[2,6,3,4,1,5]=>9
[2,6,3,4,5,1]=>9
[2,6,3,5,1,4]=>8
[2,6,3,5,4,1]=>8
[2,6,4,1,3,5]=>9
[2,6,4,1,5,3]=>11
[2,6,4,3,1,5]=>8
[2,6,4,3,5,1]=>10
[2,6,4,5,1,3]=>8
[2,6,4,5,3,1]=>8
[2,6,5,1,3,4]=>9
[2,6,5,1,4,3]=>8
[2,6,5,3,1,4]=>8
[2,6,5,3,4,1]=>10
[2,6,5,4,1,3]=>6
[2,6,5,4,3,1]=>8
[3,1,2,4,5,6]=>0
[3,1,2,4,6,5]=>0
[3,1,2,5,4,6]=>0
[3,1,2,5,6,4]=>0
[3,1,2,6,4,5]=>0
[3,1,2,6,5,4]=>0
[3,1,4,2,5,6]=>0
[3,1,4,2,6,5]=>0
[3,1,4,5,2,6]=>0
[3,1,4,5,6,2]=>0
[3,1,4,6,2,5]=>0
[3,1,4,6,5,2]=>0
[3,1,5,2,4,6]=>0
[3,1,5,2,6,4]=>0
[3,1,5,4,2,6]=>0
[3,1,5,4,6,2]=>0
[3,1,5,6,2,4]=>0
[3,1,5,6,4,2]=>0
[3,1,6,2,4,5]=>0
[3,1,6,2,5,4]=>0
[3,1,6,4,2,5]=>0
[3,1,6,4,5,2]=>0
[3,1,6,5,2,4]=>0
[3,1,6,5,4,2]=>0
[3,2,1,4,5,6]=>4
[3,2,1,4,6,5]=>4
[3,2,1,5,4,6]=>4
[3,2,1,5,6,4]=>4
[3,2,1,6,4,5]=>4
[3,2,1,6,5,4]=>5
[3,2,4,1,5,6]=>0
[3,2,4,1,6,5]=>0
[3,2,4,5,1,6]=>0
[3,2,4,5,6,1]=>0
[3,2,4,6,1,5]=>0
[3,2,4,6,5,1]=>0
[3,2,5,1,4,6]=>0
[3,2,5,1,6,4]=>0
[3,2,5,4,1,6]=>0
[3,2,5,4,6,1]=>0
[3,2,5,6,1,4]=>0
[3,2,5,6,4,1]=>0
[3,2,6,1,4,5]=>0
[3,2,6,1,5,4]=>0
[3,2,6,4,1,5]=>0
[3,2,6,4,5,1]=>0
[3,2,6,5,1,4]=>0
[3,2,6,5,4,1]=>0
[3,4,1,2,5,6]=>6
[3,4,1,2,6,5]=>7
[3,4,1,5,2,6]=>9
[3,4,1,5,6,2]=>9
[3,4,1,6,2,5]=>7
[3,4,1,6,5,2]=>8
[3,4,2,1,5,6]=>12
[3,4,2,1,6,5]=>11
[3,4,2,5,1,6]=>9
[3,4,2,5,6,1]=>9
[3,4,2,6,1,5]=>8
[3,4,2,6,5,1]=>8
[3,4,5,1,2,6]=>14
[3,4,5,1,6,2]=>16
[3,4,5,2,1,6]=>18
[3,4,5,2,6,1]=>16
[3,4,5,6,1,2]=>20
[3,4,5,6,2,1]=>22
[3,4,6,1,2,5]=>12
[3,4,6,1,5,2]=>13
[3,4,6,2,1,5]=>15
[3,4,6,2,5,1]=>13
[3,4,6,5,1,2]=>15
[3,4,6,5,2,1]=>14
[3,5,1,2,4,6]=>6
[3,5,1,2,6,4]=>7
[3,5,1,4,2,6]=>6
[3,5,1,4,6,2]=>6
[3,5,1,6,2,4]=>7
[3,5,1,6,4,2]=>8
[3,5,2,1,4,6]=>9
[3,5,2,1,6,4]=>11
[3,5,2,4,1,6]=>6
[3,5,2,4,6,1]=>6
[3,5,2,6,1,4]=>8
[3,5,2,6,4,1]=>8
[3,5,4,1,2,6]=>9
[3,5,4,1,6,2]=>11
[3,5,4,2,1,6]=>8
[3,5,4,2,6,1]=>10
[3,5,4,6,1,2]=>8
[3,5,4,6,2,1]=>8
[3,5,6,1,2,4]=>12
[3,5,6,1,4,2]=>13
[3,5,6,2,1,4]=>15
[3,5,6,2,4,1]=>13
[3,5,6,4,1,2]=>15
[3,5,6,4,2,1]=>14
[3,6,1,2,4,5]=>6
[3,6,1,2,5,4]=>6
[3,6,1,4,2,5]=>6
[3,6,1,4,5,2]=>6
[3,6,1,5,2,4]=>6
[3,6,1,5,4,2]=>6
[3,6,2,1,4,5]=>9
[3,6,2,1,5,4]=>8
[3,6,2,4,1,5]=>6
[3,6,2,4,5,1]=>6
[3,6,2,5,1,4]=>5
[3,6,2,5,4,1]=>6
[3,6,4,1,2,5]=>9
[3,6,4,1,5,2]=>11
[3,6,4,2,1,5]=>8
[3,6,4,2,5,1]=>10
[3,6,4,5,1,2]=>8
[3,6,4,5,2,1]=>8
[3,6,5,1,2,4]=>9
[3,6,5,1,4,2]=>8
[3,6,5,2,1,4]=>6
[3,6,5,2,4,1]=>8
[3,6,5,4,1,2]=>6
[3,6,5,4,2,1]=>8
[4,1,2,3,5,6]=>0
[4,1,2,3,6,5]=>0
[4,1,2,5,3,6]=>0
[4,1,2,5,6,3]=>0
[4,1,2,6,3,5]=>0
[4,1,2,6,5,3]=>0
[4,1,3,2,5,6]=>0
[4,1,3,2,6,5]=>0
[4,1,3,5,2,6]=>0
[4,1,3,5,6,2]=>0
[4,1,3,6,2,5]=>0
[4,1,3,6,5,2]=>0
[4,1,5,2,3,6]=>0
[4,1,5,2,6,3]=>0
[4,1,5,3,2,6]=>0
[4,1,5,3,6,2]=>0
[4,1,5,6,2,3]=>0
[4,1,5,6,3,2]=>0
[4,1,6,2,3,5]=>0
[4,1,6,2,5,3]=>0
[4,1,6,3,2,5]=>0
[4,1,6,3,5,2]=>0
[4,1,6,5,2,3]=>0
[4,1,6,5,3,2]=>0
[4,2,1,3,5,6]=>4
[4,2,1,3,6,5]=>4
[4,2,1,5,3,6]=>4
[4,2,1,5,6,3]=>4
[4,2,1,6,3,5]=>4
[4,2,1,6,5,3]=>5
[4,2,3,1,5,6]=>0
[4,2,3,1,6,5]=>0
[4,2,3,5,1,6]=>0
[4,2,3,5,6,1]=>0
[4,2,3,6,1,5]=>0
[4,2,3,6,5,1]=>0
[4,2,5,1,3,6]=>0
[4,2,5,1,6,3]=>0
[4,2,5,3,1,6]=>0
[4,2,5,3,6,1]=>0
[4,2,5,6,1,3]=>0
[4,2,5,6,3,1]=>0
[4,2,6,1,3,5]=>0
[4,2,6,1,5,3]=>0
[4,2,6,3,1,5]=>0
[4,2,6,3,5,1]=>0
[4,2,6,5,1,3]=>0
[4,2,6,5,3,1]=>0
[4,3,1,2,5,6]=>4
[4,3,1,2,6,5]=>4
[4,3,1,5,2,6]=>4
[4,3,1,5,6,2]=>4
[4,3,1,6,2,5]=>4
[4,3,1,6,5,2]=>5
[4,3,2,1,5,6]=>0
[4,3,2,1,6,5]=>0
[4,3,2,5,1,6]=>3
[4,3,2,5,6,1]=>3
[4,3,2,6,1,5]=>3
[4,3,2,6,5,1]=>3
[4,3,5,1,2,6]=>0
[4,3,5,1,6,2]=>0
[4,3,5,2,1,6]=>0
[4,3,5,2,6,1]=>0
[4,3,5,6,1,2]=>0
[4,3,5,6,2,1]=>0
[4,3,6,1,2,5]=>0
[4,3,6,1,5,2]=>0
[4,3,6,2,1,5]=>0
[4,3,6,2,5,1]=>0
[4,3,6,5,1,2]=>0
[4,3,6,5,2,1]=>0
[4,5,1,2,3,6]=>6
[4,5,1,2,6,3]=>7
[4,5,1,3,2,6]=>6
[4,5,1,3,6,2]=>6
[4,5,1,6,2,3]=>7
[4,5,1,6,3,2]=>8
[4,5,2,1,3,6]=>9
[4,5,2,1,6,3]=>11
[4,5,2,3,1,6]=>6
[4,5,2,3,6,1]=>6
[4,5,2,6,1,3]=>8
[4,5,2,6,3,1]=>8
[4,5,3,1,2,6]=>9
[4,5,3,1,6,2]=>11
[4,5,3,2,1,6]=>8
[4,5,3,2,6,1]=>10
[4,5,3,6,1,2]=>8
[4,5,3,6,2,1]=>8
[4,5,6,1,2,3]=>12
[4,5,6,1,3,2]=>13
[4,5,6,2,1,3]=>15
[4,5,6,2,3,1]=>13
[4,5,6,3,1,2]=>15
[4,5,6,3,2,1]=>14
[4,6,1,2,3,5]=>6
[4,6,1,2,5,3]=>6
[4,6,1,3,2,5]=>6
[4,6,1,3,5,2]=>6
[4,6,1,5,2,3]=>6
[4,6,1,5,3,2]=>6
[4,6,2,1,3,5]=>9
[4,6,2,1,5,3]=>8
[4,6,2,3,1,5]=>6
[4,6,2,3,5,1]=>6
[4,6,2,5,1,3]=>5
[4,6,2,5,3,1]=>6
[4,6,3,1,2,5]=>9
[4,6,3,1,5,2]=>8
[4,6,3,2,1,5]=>6
[4,6,3,2,5,1]=>8
[4,6,3,5,1,2]=>5
[4,6,3,5,2,1]=>6
[4,6,5,1,2,3]=>9
[4,6,5,1,3,2]=>8
[4,6,5,2,1,3]=>6
[4,6,5,2,3,1]=>8
[4,6,5,3,1,2]=>6
[4,6,5,3,2,1]=>8
[5,1,2,3,4,6]=>0
[5,1,2,3,6,4]=>0
[5,1,2,4,3,6]=>0
[5,1,2,4,6,3]=>0
[5,1,2,6,3,4]=>0
[5,1,2,6,4,3]=>0
[5,1,3,2,4,6]=>0
[5,1,3,2,6,4]=>0
[5,1,3,4,2,6]=>0
[5,1,3,4,6,2]=>0
[5,1,3,6,2,4]=>0
[5,1,3,6,4,2]=>0
[5,1,4,2,3,6]=>0
[5,1,4,2,6,3]=>0
[5,1,4,3,2,6]=>0
[5,1,4,3,6,2]=>0
[5,1,4,6,2,3]=>0
[5,1,4,6,3,2]=>0
[5,1,6,2,3,4]=>0
[5,1,6,2,4,3]=>0
[5,1,6,3,2,4]=>0
[5,1,6,3,4,2]=>0
[5,1,6,4,2,3]=>0
[5,1,6,4,3,2]=>0
[5,2,1,3,4,6]=>4
[5,2,1,3,6,4]=>4
[5,2,1,4,3,6]=>4
[5,2,1,4,6,3]=>4
[5,2,1,6,3,4]=>4
[5,2,1,6,4,3]=>5
[5,2,3,1,4,6]=>0
[5,2,3,1,6,4]=>0
[5,2,3,4,1,6]=>0
[5,2,3,4,6,1]=>0
[5,2,3,6,1,4]=>0
[5,2,3,6,4,1]=>0
[5,2,4,1,3,6]=>0
[5,2,4,1,6,3]=>0
[5,2,4,3,1,6]=>0
[5,2,4,3,6,1]=>0
[5,2,4,6,1,3]=>0
[5,2,4,6,3,1]=>0
[5,2,6,1,3,4]=>0
[5,2,6,1,4,3]=>0
[5,2,6,3,1,4]=>0
[5,2,6,3,4,1]=>0
[5,2,6,4,1,3]=>0
[5,2,6,4,3,1]=>0
[5,3,1,2,4,6]=>4
[5,3,1,2,6,4]=>4
[5,3,1,4,2,6]=>4
[5,3,1,4,6,2]=>4
[5,3,1,6,2,4]=>4
[5,3,1,6,4,2]=>5
[5,3,2,1,4,6]=>0
[5,3,2,1,6,4]=>0
[5,3,2,4,1,6]=>3
[5,3,2,4,6,1]=>3
[5,3,2,6,1,4]=>3
[5,3,2,6,4,1]=>3
[5,3,4,1,2,6]=>0
[5,3,4,1,6,2]=>0
[5,3,4,2,1,6]=>0
[5,3,4,2,6,1]=>0
[5,3,4,6,1,2]=>0
[5,3,4,6,2,1]=>0
[5,3,6,1,2,4]=>0
[5,3,6,1,4,2]=>0
[5,3,6,2,1,4]=>0
[5,3,6,2,4,1]=>0
[5,3,6,4,1,2]=>0
[5,3,6,4,2,1]=>0
[5,4,1,2,3,6]=>4
[5,4,1,2,6,3]=>4
[5,4,1,3,2,6]=>3
[5,4,1,3,6,2]=>3
[5,4,1,6,2,3]=>4
[5,4,1,6,3,2]=>3
[5,4,2,1,3,6]=>0
[5,4,2,1,6,3]=>0
[5,4,2,3,1,6]=>3
[5,4,2,3,6,1]=>3
[5,4,2,6,1,3]=>3
[5,4,2,6,3,1]=>3
[5,4,3,1,2,6]=>0
[5,4,3,1,6,2]=>0
[5,4,3,2,1,6]=>2
[5,4,3,2,6,1]=>0
[5,4,3,6,1,2]=>3
[5,4,3,6,2,1]=>2
[5,4,6,1,2,3]=>0
[5,4,6,1,3,2]=>0
[5,4,6,2,1,3]=>0
[5,4,6,2,3,1]=>0
[5,4,6,3,1,2]=>0
[5,4,6,3,2,1]=>0
[5,6,1,2,3,4]=>6
[5,6,1,2,4,3]=>6
[5,6,1,3,2,4]=>6
[5,6,1,3,4,2]=>6
[5,6,1,4,2,3]=>6
[5,6,1,4,3,2]=>6
[5,6,2,1,3,4]=>9
[5,6,2,1,4,3]=>8
[5,6,2,3,1,4]=>6
[5,6,2,3,4,1]=>6
[5,6,2,4,1,3]=>5
[5,6,2,4,3,1]=>6
[5,6,3,1,2,4]=>9
[5,6,3,1,4,2]=>8
[5,6,3,2,1,4]=>6
[5,6,3,2,4,1]=>8
[5,6,3,4,1,2]=>5
[5,6,3,4,2,1]=>6
[5,6,4,1,2,3]=>9
[5,6,4,1,3,2]=>8
[5,6,4,2,1,3]=>6
[5,6,4,2,3,1]=>8
[5,6,4,3,1,2]=>6
[5,6,4,3,2,1]=>8
[6,1,2,3,4,5]=>0
[6,1,2,3,5,4]=>0
[6,1,2,4,3,5]=>0
[6,1,2,4,5,3]=>0
[6,1,2,5,3,4]=>0
[6,1,2,5,4,3]=>0
[6,1,3,2,4,5]=>0
[6,1,3,2,5,4]=>0
[6,1,3,4,2,5]=>0
[6,1,3,4,5,2]=>0
[6,1,3,5,2,4]=>0
[6,1,3,5,4,2]=>0
[6,1,4,2,3,5]=>0
[6,1,4,2,5,3]=>0
[6,1,4,3,2,5]=>0
[6,1,4,3,5,2]=>0
[6,1,4,5,2,3]=>0
[6,1,4,5,3,2]=>0
[6,1,5,2,3,4]=>0
[6,1,5,2,4,3]=>0
[6,1,5,3,2,4]=>0
[6,1,5,3,4,2]=>0
[6,1,5,4,2,3]=>0
[6,1,5,4,3,2]=>0
[6,2,1,3,4,5]=>4
[6,2,1,3,5,4]=>4
[6,2,1,4,3,5]=>4
[6,2,1,4,5,3]=>4
[6,2,1,5,3,4]=>4
[6,2,1,5,4,3]=>3
[6,2,3,1,4,5]=>0
[6,2,3,1,5,4]=>0
[6,2,3,4,1,5]=>0
[6,2,3,4,5,1]=>0
[6,2,3,5,1,4]=>0
[6,2,3,5,4,1]=>0
[6,2,4,1,3,5]=>0
[6,2,4,1,5,3]=>0
[6,2,4,3,1,5]=>0
[6,2,4,3,5,1]=>0
[6,2,4,5,1,3]=>0
[6,2,4,5,3,1]=>0
[6,2,5,1,3,4]=>0
[6,2,5,1,4,3]=>0
[6,2,5,3,1,4]=>0
[6,2,5,3,4,1]=>0
[6,2,5,4,1,3]=>0
[6,2,5,4,3,1]=>0
[6,3,1,2,4,5]=>4
[6,3,1,2,5,4]=>4
[6,3,1,4,2,5]=>4
[6,3,1,4,5,2]=>4
[6,3,1,5,2,4]=>4
[6,3,1,5,4,2]=>3
[6,3,2,1,4,5]=>0
[6,3,2,1,5,4]=>0
[6,3,2,4,1,5]=>3
[6,3,2,4,5,1]=>3
[6,3,2,5,1,4]=>3
[6,3,2,5,4,1]=>3
[6,3,4,1,2,5]=>0
[6,3,4,1,5,2]=>0
[6,3,4,2,1,5]=>0
[6,3,4,2,5,1]=>0
[6,3,4,5,1,2]=>0
[6,3,4,5,2,1]=>0
[6,3,5,1,2,4]=>0
[6,3,5,1,4,2]=>0
[6,3,5,2,1,4]=>0
[6,3,5,2,4,1]=>0
[6,3,5,4,1,2]=>0
[6,3,5,4,2,1]=>0
[6,4,1,2,3,5]=>4
[6,4,1,2,5,3]=>4
[6,4,1,3,2,5]=>3
[6,4,1,3,5,2]=>3
[6,4,1,5,2,3]=>4
[6,4,1,5,3,2]=>3
[6,4,2,1,3,5]=>0
[6,4,2,1,5,3]=>0
[6,4,2,3,1,5]=>3
[6,4,2,3,5,1]=>3
[6,4,2,5,1,3]=>3
[6,4,2,5,3,1]=>3
[6,4,3,1,2,5]=>0
[6,4,3,1,5,2]=>0
[6,4,3,2,1,5]=>2
[6,4,3,2,5,1]=>0
[6,4,3,5,1,2]=>3
[6,4,3,5,2,1]=>2
[6,4,5,1,2,3]=>0
[6,4,5,1,3,2]=>0
[6,4,5,2,1,3]=>0
[6,4,5,2,3,1]=>0
[6,4,5,3,1,2]=>0
[6,4,5,3,2,1]=>0
[6,5,1,2,3,4]=>4
[6,5,1,2,4,3]=>3
[6,5,1,3,2,4]=>3
[6,5,1,3,4,2]=>3
[6,5,1,4,2,3]=>3
[6,5,1,4,3,2]=>2
[6,5,2,1,3,4]=>0
[6,5,2,1,4,3]=>0
[6,5,2,3,1,4]=>3
[6,5,2,3,4,1]=>3
[6,5,2,4,1,3]=>3
[6,5,2,4,3,1]=>2
[6,5,3,1,2,4]=>0
[6,5,3,1,4,2]=>0
[6,5,3,2,1,4]=>2
[6,5,3,2,4,1]=>0
[6,5,3,4,1,2]=>3
[6,5,3,4,2,1]=>2
[6,5,4,1,2,3]=>0
[6,5,4,1,3,2]=>0
[6,5,4,2,1,3]=>2
[6,5,4,2,3,1]=>0
[6,5,4,3,1,2]=>2
[6,5,4,3,2,1]=>0
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Description
Eigenvalues of the random-to-random operator acting on the regular representation.
This statistic is defined for a permutation $w$ as:
$$ \left[\binom{\ell(w) + 1}{2} + \operatorname{diag}\left(Q(w)\right)\right] - \left[\binom{\ell(u) + 1}{2} + \operatorname{diag}\left(Q(u)\right)\right] $$
where:
This statistic is defined for a permutation $w$ as:
$$ \left[\binom{\ell(w) + 1}{2} + \operatorname{diag}\left(Q(w)\right)\right] - \left[\binom{\ell(u) + 1}{2} + \operatorname{diag}\left(Q(u)\right)\right] $$
where:
- $u$ is the longest suffix of $w$ (viewed as a word) whose first ascent is even;
- $\ell(w)$ is the size of the permutation $w$ (equivalently, the length of the word $w$);
- $Q(w), Q(u)$ denote the recording tableaux of $w, u$ under the RSK correspondence;
- $\operatorname{diag}(\lambda)$ denotes the diagonal index (or content) of an integer partition $\lambda$;
- and $\operatorname{diag}(T)$ of a tableau $T$ denotes the diagonal index of the partition given by the shape of $T$.
References
[1] Dieker, A. B., Saliola, F. Spectral analysis of random-to-random Markov chains arXiv:1509.08580
Code
def is_desarrangement_word(w):
r"""
EXAMPLES::
sage: for n in range(4):
....: for perm in Permutations(n):
....: print "%s => %s" % (perm, is_desarrangement_word(perm))
[] => True
[1] => False
[1, 2] => False
[2, 1] => True
[1, 2, 3] => False
[1, 3, 2] => False
[2, 1, 3] => True
[2, 3, 1] => False
[3, 1, 2] => True
[3, 2, 1] => False
"""
X = [k for k in range(len(w)-1) if w[k] <= w[k+1]]
if X:
return X[0] % 2 == 1
else:
return len(w) % 2 == 0
def desarrangement_factorization(w):
r"""
EXAMPLES::
sage: for n in range(4):
....: for perm in Permutations(n):
....: print "%s => %s" % (perm, desarrangement_factorization(perm))
[] => ([], [])
[1] => ([1], [])
[1, 2] => ([1, 2], [])
[2, 1] => ([], [2, 1])
[1, 2, 3] => ([1, 2, 3], [])
[1, 3, 2] => ([1], [3, 2])
[2, 1, 3] => ([], [2, 1, 3])
[2, 3, 1] => ([2], [3, 1])
[3, 1, 2] => ([], [3, 1, 2])
[3, 2, 1] => ([3], [2, 1])
"""
for i in range(len(w) + 1):
u = Word(w[i:]).standard_permutation()
if is_desarrangement_word(u):
return w[:i], w[i:]
def diagonal_index_of_partition(la):
return sum((j - i) for (i, j) in la.cells())
def binomial_shifted_diagonal_index_of_partition(partition):
return binomial(partition.size() + 1, 2) + diagonal_index_of_partition(partition)
def statistic(w):
r"""
EXAMPLES::
sage: for n in range(5):
....: for perm in Permutations(n):
....: print "%s => %s" % (perm, statistic(perm))
[] => 0
[1] => 1
[1, 2] => 4
[2, 1] => 0
[1, 2, 3] => 9
[1, 3, 2] => 4
[2, 1, 3] => 0
[2, 3, 1] => 4
[3, 1, 2] => 0
[3, 2, 1] => 1
"""
Qw = RSK(w)[1]
_, u = desarrangement_factorization(w)
Qu = RSK(u)[1]
Qw_index = binomial_shifted_diagonal_index_of_partition(Qw.shape())
Qu_index = binomial_shifted_diagonal_index_of_partition(Qu.shape())
return Qw_index - Qu_index
Created
May 23, 2016 at 21:58 by Franco Saliola
Updated
Jan 13, 2018 at 15:45 by Martin Rubey
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