Identifier
- St001911: Permutations ⟶ ℤ
Values
=>
[1]=>0
[1,2]=>0
[2,1]=>0
[1,2,3]=>0
[1,3,2]=>1
[2,1,3]=>1
[2,3,1]=>0
[3,1,2]=>0
[3,2,1]=>1
[1,2,3,4]=>0
[1,2,4,3]=>2
[1,3,2,4]=>3
[1,3,4,2]=>1
[1,4,2,3]=>2
[1,4,3,2]=>4
[2,1,3,4]=>2
[2,1,4,3]=>4
[2,3,1,4]=>2
[2,3,4,1]=>0
[2,4,1,3]=>1
[2,4,3,1]=>3
[3,1,2,4]=>1
[3,1,4,2]=>3
[3,2,1,4]=>4
[3,2,4,1]=>2
[3,4,1,2]=>0
[3,4,2,1]=>2
[4,1,2,3]=>0
[4,1,3,2]=>2
[4,2,1,3]=>3
[4,2,3,1]=>1
[4,3,1,2]=>2
[4,3,2,1]=>4
[1,2,3,4,5]=>0
[1,2,3,5,4]=>3
[1,2,4,3,5]=>5
[1,2,4,5,3]=>2
[1,2,5,3,4]=>4
[1,2,5,4,3]=>7
[1,3,2,4,5]=>5
[1,3,2,5,4]=>8
[1,3,4,2,5]=>4
[1,3,4,5,2]=>1
[1,3,5,2,4]=>3
[1,3,5,4,2]=>6
[1,4,2,3,5]=>4
[1,4,2,5,3]=>7
[1,4,3,2,5]=>9
[1,4,3,5,2]=>6
[1,4,5,2,3]=>2
[1,4,5,3,2]=>5
[1,5,2,3,4]=>3
[1,5,2,4,3]=>6
[1,5,3,2,4]=>8
[1,5,3,4,2]=>5
[1,5,4,2,3]=>7
[1,5,4,3,2]=>10
[2,1,3,4,5]=>3
[2,1,3,5,4]=>6
[2,1,4,3,5]=>8
[2,1,4,5,3]=>5
[2,1,5,3,4]=>7
[2,1,5,4,3]=>10
[2,3,1,4,5]=>4
[2,3,1,5,4]=>7
[2,3,4,1,5]=>3
[2,3,4,5,1]=>0
[2,3,5,1,4]=>2
[2,3,5,4,1]=>5
[2,4,1,3,5]=>3
[2,4,1,5,3]=>6
[2,4,3,1,5]=>8
[2,4,3,5,1]=>5
[2,4,5,1,3]=>1
[2,4,5,3,1]=>4
[2,5,1,3,4]=>2
[2,5,1,4,3]=>5
[2,5,3,1,4]=>7
[2,5,3,4,1]=>4
[2,5,4,1,3]=>6
[2,5,4,3,1]=>9
[3,1,2,4,5]=>2
[3,1,2,5,4]=>5
[3,1,4,2,5]=>7
[3,1,4,5,2]=>4
[3,1,5,2,4]=>6
[3,1,5,4,2]=>9
[3,2,1,4,5]=>7
[3,2,1,5,4]=>10
[3,2,4,1,5]=>6
[3,2,4,5,1]=>3
[3,2,5,1,4]=>5
[3,2,5,4,1]=>8
[3,4,1,2,5]=>2
[3,4,1,5,2]=>5
[3,4,2,1,5]=>7
[3,4,2,5,1]=>4
[3,4,5,1,2]=>0
[3,4,5,2,1]=>3
[3,5,1,2,4]=>1
[3,5,1,4,2]=>4
[3,5,2,1,4]=>6
[3,5,2,4,1]=>3
[3,5,4,1,2]=>5
[3,5,4,2,1]=>8
[4,1,2,3,5]=>1
[4,1,2,5,3]=>4
[4,1,3,2,5]=>6
[4,1,3,5,2]=>3
[4,1,5,2,3]=>5
[4,1,5,3,2]=>8
[4,2,1,3,5]=>6
[4,2,1,5,3]=>9
[4,2,3,1,5]=>5
[4,2,3,5,1]=>2
[4,2,5,1,3]=>4
[4,2,5,3,1]=>7
[4,3,1,2,5]=>5
[4,3,1,5,2]=>8
[4,3,2,1,5]=>10
[4,3,2,5,1]=>7
[4,3,5,1,2]=>3
[4,3,5,2,1]=>6
[4,5,1,2,3]=>0
[4,5,1,3,2]=>3
[4,5,2,1,3]=>5
[4,5,2,3,1]=>2
[4,5,3,1,2]=>4
[4,5,3,2,1]=>7
[5,1,2,3,4]=>0
[5,1,2,4,3]=>3
[5,1,3,2,4]=>5
[5,1,3,4,2]=>2
[5,1,4,2,3]=>4
[5,1,4,3,2]=>7
[5,2,1,3,4]=>5
[5,2,1,4,3]=>8
[5,2,3,1,4]=>4
[5,2,3,4,1]=>1
[5,2,4,1,3]=>3
[5,2,4,3,1]=>6
[5,3,1,2,4]=>4
[5,3,1,4,2]=>7
[5,3,2,1,4]=>9
[5,3,2,4,1]=>6
[5,3,4,1,2]=>2
[5,3,4,2,1]=>5
[5,4,1,2,3]=>3
[5,4,1,3,2]=>6
[5,4,2,1,3]=>8
[5,4,2,3,1]=>5
[5,4,3,1,2]=>7
[5,4,3,2,1]=>10
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>4
[1,2,3,5,4,6]=>7
[1,2,3,5,6,4]=>3
[1,2,3,6,4,5]=>6
[1,2,3,6,5,4]=>10
[1,2,4,3,5,6]=>8
[1,2,4,3,6,5]=>12
[1,2,4,5,3,6]=>6
[1,2,4,5,6,3]=>2
[1,2,4,6,3,5]=>5
[1,2,4,6,5,3]=>9
[1,2,5,3,4,6]=>7
[1,2,5,3,6,4]=>11
[1,2,5,4,3,6]=>14
[1,2,5,4,6,3]=>10
[1,2,5,6,3,4]=>4
[1,2,5,6,4,3]=>8
[1,2,6,3,4,5]=>6
[1,2,6,3,5,4]=>10
[1,2,6,4,3,5]=>13
[1,2,6,4,5,3]=>9
[1,2,6,5,3,4]=>12
[1,2,6,5,4,3]=>16
[1,3,2,4,5,6]=>7
[1,3,2,4,6,5]=>11
[1,3,2,5,4,6]=>14
[1,3,2,5,6,4]=>10
[1,3,2,6,4,5]=>13
[1,3,2,6,5,4]=>17
[1,3,4,2,5,6]=>7
[1,3,4,2,6,5]=>11
[1,3,4,5,2,6]=>5
[1,3,4,5,6,2]=>1
[1,3,4,6,2,5]=>4
[1,3,4,6,5,2]=>8
[1,3,5,2,4,6]=>6
[1,3,5,2,6,4]=>10
[1,3,5,4,2,6]=>13
[1,3,5,4,6,2]=>9
[1,3,5,6,2,4]=>3
[1,3,5,6,4,2]=>7
[1,3,6,2,4,5]=>5
[1,3,6,2,5,4]=>9
[1,3,6,4,2,5]=>12
[1,3,6,4,5,2]=>8
[1,3,6,5,2,4]=>11
[1,3,6,5,4,2]=>15
[1,4,2,3,5,6]=>6
[1,4,2,3,6,5]=>10
[1,4,2,5,3,6]=>13
[1,4,2,5,6,3]=>9
[1,4,2,6,3,5]=>12
[1,4,2,6,5,3]=>16
[1,4,3,2,5,6]=>14
[1,4,3,2,6,5]=>18
[1,4,3,5,2,6]=>12
[1,4,3,5,6,2]=>8
[1,4,3,6,2,5]=>11
[1,4,3,6,5,2]=>15
[1,4,5,2,3,6]=>5
[1,4,5,2,6,3]=>9
[1,4,5,3,2,6]=>12
[1,4,5,3,6,2]=>8
[1,4,5,6,2,3]=>2
[1,4,5,6,3,2]=>6
[1,4,6,2,3,5]=>4
[1,4,6,2,5,3]=>8
[1,4,6,3,2,5]=>11
[1,4,6,3,5,2]=>7
[1,4,6,5,2,3]=>10
[1,4,6,5,3,2]=>14
[1,5,2,3,4,6]=>5
[1,5,2,3,6,4]=>9
[1,5,2,4,3,6]=>12
[1,5,2,4,6,3]=>8
[1,5,2,6,3,4]=>11
[1,5,2,6,4,3]=>15
[1,5,3,2,4,6]=>13
[1,5,3,2,6,4]=>17
[1,5,3,4,2,6]=>11
[1,5,3,4,6,2]=>7
[1,5,3,6,2,4]=>10
[1,5,3,6,4,2]=>14
[1,5,4,2,3,6]=>12
[1,5,4,2,6,3]=>16
[1,5,4,3,2,6]=>19
[1,5,4,3,6,2]=>15
[1,5,4,6,2,3]=>9
[1,5,4,6,3,2]=>13
[1,5,6,2,3,4]=>3
[1,5,6,2,4,3]=>7
[1,5,6,3,2,4]=>10
[1,5,6,3,4,2]=>6
[1,5,6,4,2,3]=>9
[1,5,6,4,3,2]=>13
[1,6,2,3,4,5]=>4
[1,6,2,3,5,4]=>8
[1,6,2,4,3,5]=>11
[1,6,2,4,5,3]=>7
[1,6,2,5,3,4]=>10
[1,6,2,5,4,3]=>14
[1,6,3,2,4,5]=>12
[1,6,3,2,5,4]=>16
[1,6,3,4,2,5]=>10
[1,6,3,4,5,2]=>6
[1,6,3,5,2,4]=>9
[1,6,3,5,4,2]=>13
[1,6,4,2,3,5]=>11
[1,6,4,2,5,3]=>15
[1,6,4,3,2,5]=>18
[1,6,4,3,5,2]=>14
[1,6,4,5,2,3]=>8
[1,6,4,5,3,2]=>12
[1,6,5,2,3,4]=>10
[1,6,5,2,4,3]=>14
[1,6,5,3,2,4]=>17
[1,6,5,3,4,2]=>13
[1,6,5,4,2,3]=>16
[1,6,5,4,3,2]=>20
[2,1,3,4,5,6]=>4
[2,1,3,4,6,5]=>8
[2,1,3,5,4,6]=>11
[2,1,3,5,6,4]=>7
[2,1,3,6,4,5]=>10
[2,1,3,6,5,4]=>14
[2,1,4,3,5,6]=>12
[2,1,4,3,6,5]=>16
[2,1,4,5,3,6]=>10
[2,1,4,5,6,3]=>6
[2,1,4,6,3,5]=>9
[2,1,4,6,5,3]=>13
[2,1,5,3,4,6]=>11
[2,1,5,3,6,4]=>15
[2,1,5,4,3,6]=>18
[2,1,5,4,6,3]=>14
[2,1,5,6,3,4]=>8
[2,1,5,6,4,3]=>12
[2,1,6,3,4,5]=>10
[2,1,6,3,5,4]=>14
[2,1,6,4,3,5]=>17
[2,1,6,4,5,3]=>13
[2,1,6,5,3,4]=>16
[2,1,6,5,4,3]=>20
[2,3,1,4,5,6]=>6
[2,3,1,4,6,5]=>10
[2,3,1,5,4,6]=>13
[2,3,1,5,6,4]=>9
[2,3,1,6,4,5]=>12
[2,3,1,6,5,4]=>16
[2,3,4,1,5,6]=>6
[2,3,4,1,6,5]=>10
[2,3,4,5,1,6]=>4
[2,3,4,5,6,1]=>0
[2,3,4,6,1,5]=>3
[2,3,4,6,5,1]=>7
[2,3,5,1,4,6]=>5
[2,3,5,1,6,4]=>9
[2,3,5,4,1,6]=>12
[2,3,5,4,6,1]=>8
[2,3,5,6,1,4]=>2
[2,3,5,6,4,1]=>6
[2,3,6,1,4,5]=>4
[2,3,6,1,5,4]=>8
[2,3,6,4,1,5]=>11
[2,3,6,4,5,1]=>7
[2,3,6,5,1,4]=>10
[2,3,6,5,4,1]=>14
[2,4,1,3,5,6]=>5
[2,4,1,3,6,5]=>9
[2,4,1,5,3,6]=>12
[2,4,1,5,6,3]=>8
[2,4,1,6,3,5]=>11
[2,4,1,6,5,3]=>15
[2,4,3,1,5,6]=>13
[2,4,3,1,6,5]=>17
[2,4,3,5,1,6]=>11
[2,4,3,5,6,1]=>7
[2,4,3,6,1,5]=>10
[2,4,3,6,5,1]=>14
[2,4,5,1,3,6]=>4
[2,4,5,1,6,3]=>8
[2,4,5,3,1,6]=>11
[2,4,5,3,6,1]=>7
[2,4,5,6,1,3]=>1
[2,4,5,6,3,1]=>5
[2,4,6,1,3,5]=>3
[2,4,6,1,5,3]=>7
[2,4,6,3,1,5]=>10
[2,4,6,3,5,1]=>6
[2,4,6,5,1,3]=>9
[2,4,6,5,3,1]=>13
[2,5,1,3,4,6]=>4
[2,5,1,3,6,4]=>8
[2,5,1,4,3,6]=>11
[2,5,1,4,6,3]=>7
[2,5,1,6,3,4]=>10
[2,5,1,6,4,3]=>14
[2,5,3,1,4,6]=>12
[2,5,3,1,6,4]=>16
[2,5,3,4,1,6]=>10
[2,5,3,4,6,1]=>6
[2,5,3,6,1,4]=>9
[2,5,3,6,4,1]=>13
[2,5,4,1,3,6]=>11
[2,5,4,1,6,3]=>15
[2,5,4,3,1,6]=>18
[2,5,4,3,6,1]=>14
[2,5,4,6,1,3]=>8
[2,5,4,6,3,1]=>12
[2,5,6,1,3,4]=>2
[2,5,6,1,4,3]=>6
[2,5,6,3,1,4]=>9
[2,5,6,3,4,1]=>5
[2,5,6,4,1,3]=>8
[2,5,6,4,3,1]=>12
[2,6,1,3,4,5]=>3
[2,6,1,3,5,4]=>7
[2,6,1,4,3,5]=>10
[2,6,1,4,5,3]=>6
[2,6,1,5,3,4]=>9
[2,6,1,5,4,3]=>13
[2,6,3,1,4,5]=>11
[2,6,3,1,5,4]=>15
[2,6,3,4,1,5]=>9
[2,6,3,4,5,1]=>5
[2,6,3,5,1,4]=>8
[2,6,3,5,4,1]=>12
[2,6,4,1,3,5]=>10
[2,6,4,1,5,3]=>14
[2,6,4,3,1,5]=>17
[2,6,4,3,5,1]=>13
[2,6,4,5,1,3]=>7
[2,6,4,5,3,1]=>11
[2,6,5,1,3,4]=>9
[2,6,5,1,4,3]=>13
[2,6,5,3,1,4]=>16
[2,6,5,3,4,1]=>12
[2,6,5,4,1,3]=>15
[2,6,5,4,3,1]=>19
[3,1,2,4,5,6]=>3
[3,1,2,4,6,5]=>7
[3,1,2,5,4,6]=>10
[3,1,2,5,6,4]=>6
[3,1,2,6,4,5]=>9
[3,1,2,6,5,4]=>13
[3,1,4,2,5,6]=>11
[3,1,4,2,6,5]=>15
[3,1,4,5,2,6]=>9
[3,1,4,5,6,2]=>5
[3,1,4,6,2,5]=>8
[3,1,4,6,5,2]=>12
[3,1,5,2,4,6]=>10
[3,1,5,2,6,4]=>14
[3,1,5,4,2,6]=>17
[3,1,5,4,6,2]=>13
[3,1,5,6,2,4]=>7
[3,1,5,6,4,2]=>11
[3,1,6,2,4,5]=>9
[3,1,6,2,5,4]=>13
[3,1,6,4,2,5]=>16
[3,1,6,4,5,2]=>12
[3,1,6,5,2,4]=>15
[3,1,6,5,4,2]=>19
[3,2,1,4,5,6]=>10
[3,2,1,4,6,5]=>14
[3,2,1,5,4,6]=>17
[3,2,1,5,6,4]=>13
[3,2,1,6,4,5]=>16
[3,2,1,6,5,4]=>20
[3,2,4,1,5,6]=>10
[3,2,4,1,6,5]=>14
[3,2,4,5,1,6]=>8
[3,2,4,5,6,1]=>4
[3,2,4,6,1,5]=>7
[3,2,4,6,5,1]=>11
[3,2,5,1,4,6]=>9
[3,2,5,1,6,4]=>13
[3,2,5,4,1,6]=>16
[3,2,5,4,6,1]=>12
[3,2,5,6,1,4]=>6
[3,2,5,6,4,1]=>10
[3,2,6,1,4,5]=>8
[3,2,6,1,5,4]=>12
[3,2,6,4,1,5]=>15
[3,2,6,4,5,1]=>11
[3,2,6,5,1,4]=>14
[3,2,6,5,4,1]=>18
[3,4,1,2,5,6]=>4
[3,4,1,2,6,5]=>8
[3,4,1,5,2,6]=>11
[3,4,1,5,6,2]=>7
[3,4,1,6,2,5]=>10
[3,4,1,6,5,2]=>14
[3,4,2,1,5,6]=>12
[3,4,2,1,6,5]=>16
[3,4,2,5,1,6]=>10
[3,4,2,5,6,1]=>6
[3,4,2,6,1,5]=>9
[3,4,2,6,5,1]=>13
[3,4,5,1,2,6]=>3
[3,4,5,1,6,2]=>7
[3,4,5,2,1,6]=>10
[3,4,5,2,6,1]=>6
[3,4,5,6,1,2]=>0
[3,4,5,6,2,1]=>4
[3,4,6,1,2,5]=>2
[3,4,6,1,5,2]=>6
[3,4,6,2,1,5]=>9
[3,4,6,2,5,1]=>5
[3,4,6,5,1,2]=>8
[3,4,6,5,2,1]=>12
[3,5,1,2,4,6]=>3
[3,5,1,2,6,4]=>7
[3,5,1,4,2,6]=>10
[3,5,1,4,6,2]=>6
[3,5,1,6,2,4]=>9
[3,5,1,6,4,2]=>13
[3,5,2,1,4,6]=>11
[3,5,2,1,6,4]=>15
[3,5,2,4,1,6]=>9
[3,5,2,4,6,1]=>5
[3,5,2,6,1,4]=>8
[3,5,2,6,4,1]=>12
[3,5,4,1,2,6]=>10
[3,5,4,1,6,2]=>14
[3,5,4,2,1,6]=>17
[3,5,4,2,6,1]=>13
[3,5,4,6,1,2]=>7
[3,5,4,6,2,1]=>11
[3,5,6,1,2,4]=>1
[3,5,6,1,4,2]=>5
[3,5,6,2,1,4]=>8
[3,5,6,2,4,1]=>4
[3,5,6,4,1,2]=>7
[3,5,6,4,2,1]=>11
[3,6,1,2,4,5]=>2
[3,6,1,2,5,4]=>6
[3,6,1,4,2,5]=>9
[3,6,1,4,5,2]=>5
[3,6,1,5,2,4]=>8
[3,6,1,5,4,2]=>12
[3,6,2,1,4,5]=>10
[3,6,2,1,5,4]=>14
[3,6,2,4,1,5]=>8
[3,6,2,4,5,1]=>4
[3,6,2,5,1,4]=>7
[3,6,2,5,4,1]=>11
[3,6,4,1,2,5]=>9
[3,6,4,1,5,2]=>13
[3,6,4,2,1,5]=>16
[3,6,4,2,5,1]=>12
[3,6,4,5,1,2]=>6
[3,6,4,5,2,1]=>10
[3,6,5,1,2,4]=>8
[3,6,5,1,4,2]=>12
[3,6,5,2,1,4]=>15
[3,6,5,2,4,1]=>11
[3,6,5,4,1,2]=>14
[3,6,5,4,2,1]=>18
[4,1,2,3,5,6]=>2
[4,1,2,3,6,5]=>6
[4,1,2,5,3,6]=>9
[4,1,2,5,6,3]=>5
[4,1,2,6,3,5]=>8
[4,1,2,6,5,3]=>12
[4,1,3,2,5,6]=>10
[4,1,3,2,6,5]=>14
[4,1,3,5,2,6]=>8
[4,1,3,5,6,2]=>4
[4,1,3,6,2,5]=>7
[4,1,3,6,5,2]=>11
[4,1,5,2,3,6]=>9
[4,1,5,2,6,3]=>13
[4,1,5,3,2,6]=>16
[4,1,5,3,6,2]=>12
[4,1,5,6,2,3]=>6
[4,1,5,6,3,2]=>10
[4,1,6,2,3,5]=>8
[4,1,6,2,5,3]=>12
[4,1,6,3,2,5]=>15
[4,1,6,3,5,2]=>11
[4,1,6,5,2,3]=>14
[4,1,6,5,3,2]=>18
[4,2,1,3,5,6]=>9
[4,2,1,3,6,5]=>13
[4,2,1,5,3,6]=>16
[4,2,1,5,6,3]=>12
[4,2,1,6,3,5]=>15
[4,2,1,6,5,3]=>19
[4,2,3,1,5,6]=>9
[4,2,3,1,6,5]=>13
[4,2,3,5,1,6]=>7
[4,2,3,5,6,1]=>3
[4,2,3,6,1,5]=>6
[4,2,3,6,5,1]=>10
[4,2,5,1,3,6]=>8
[4,2,5,1,6,3]=>12
[4,2,5,3,1,6]=>15
[4,2,5,3,6,1]=>11
[4,2,5,6,1,3]=>5
[4,2,5,6,3,1]=>9
[4,2,6,1,3,5]=>7
[4,2,6,1,5,3]=>11
[4,2,6,3,1,5]=>14
[4,2,6,3,5,1]=>10
[4,2,6,5,1,3]=>13
[4,2,6,5,3,1]=>17
[4,3,1,2,5,6]=>8
[4,3,1,2,6,5]=>12
[4,3,1,5,2,6]=>15
[4,3,1,5,6,2]=>11
[4,3,1,6,2,5]=>14
[4,3,1,6,5,2]=>18
[4,3,2,1,5,6]=>16
[4,3,2,1,6,5]=>20
[4,3,2,5,1,6]=>14
[4,3,2,5,6,1]=>10
[4,3,2,6,1,5]=>13
[4,3,2,6,5,1]=>17
[4,3,5,1,2,6]=>7
[4,3,5,1,6,2]=>11
[4,3,5,2,1,6]=>14
[4,3,5,2,6,1]=>10
[4,3,5,6,1,2]=>4
[4,3,5,6,2,1]=>8
[4,3,6,1,2,5]=>6
[4,3,6,1,5,2]=>10
[4,3,6,2,1,5]=>13
[4,3,6,2,5,1]=>9
[4,3,6,5,1,2]=>12
[4,3,6,5,2,1]=>16
[4,5,1,2,3,6]=>2
[4,5,1,2,6,3]=>6
[4,5,1,3,2,6]=>9
[4,5,1,3,6,2]=>5
[4,5,1,6,2,3]=>8
[4,5,1,6,3,2]=>12
[4,5,2,1,3,6]=>10
[4,5,2,1,6,3]=>14
[4,5,2,3,1,6]=>8
[4,5,2,3,6,1]=>4
[4,5,2,6,1,3]=>7
[4,5,2,6,3,1]=>11
[4,5,3,1,2,6]=>9
[4,5,3,1,6,2]=>13
[4,5,3,2,1,6]=>16
[4,5,3,2,6,1]=>12
[4,5,3,6,1,2]=>6
[4,5,3,6,2,1]=>10
[4,5,6,1,2,3]=>0
[4,5,6,1,3,2]=>4
[4,5,6,2,1,3]=>7
[4,5,6,2,3,1]=>3
[4,5,6,3,1,2]=>6
[4,5,6,3,2,1]=>10
[4,6,1,2,3,5]=>1
[4,6,1,2,5,3]=>5
[4,6,1,3,2,5]=>8
[4,6,1,3,5,2]=>4
[4,6,1,5,2,3]=>7
[4,6,1,5,3,2]=>11
[4,6,2,1,3,5]=>9
[4,6,2,1,5,3]=>13
[4,6,2,3,1,5]=>7
[4,6,2,3,5,1]=>3
[4,6,2,5,1,3]=>6
[4,6,2,5,3,1]=>10
[4,6,3,1,2,5]=>8
[4,6,3,1,5,2]=>12
[4,6,3,2,1,5]=>15
[4,6,3,2,5,1]=>11
[4,6,3,5,1,2]=>5
[4,6,3,5,2,1]=>9
[4,6,5,1,2,3]=>7
[4,6,5,1,3,2]=>11
[4,6,5,2,1,3]=>14
[4,6,5,2,3,1]=>10
[4,6,5,3,1,2]=>13
[4,6,5,3,2,1]=>17
[5,1,2,3,4,6]=>1
[5,1,2,3,6,4]=>5
[5,1,2,4,3,6]=>8
[5,1,2,4,6,3]=>4
[5,1,2,6,3,4]=>7
[5,1,2,6,4,3]=>11
[5,1,3,2,4,6]=>9
[5,1,3,2,6,4]=>13
[5,1,3,4,2,6]=>7
[5,1,3,4,6,2]=>3
[5,1,3,6,2,4]=>6
[5,1,3,6,4,2]=>10
[5,1,4,2,3,6]=>8
[5,1,4,2,6,3]=>12
[5,1,4,3,2,6]=>15
[5,1,4,3,6,2]=>11
[5,1,4,6,2,3]=>5
[5,1,4,6,3,2]=>9
[5,1,6,2,3,4]=>7
[5,1,6,2,4,3]=>11
[5,1,6,3,2,4]=>14
[5,1,6,3,4,2]=>10
[5,1,6,4,2,3]=>13
[5,1,6,4,3,2]=>17
[5,2,1,3,4,6]=>8
[5,2,1,3,6,4]=>12
[5,2,1,4,3,6]=>15
[5,2,1,4,6,3]=>11
[5,2,1,6,3,4]=>14
[5,2,1,6,4,3]=>18
[5,2,3,1,4,6]=>8
[5,2,3,1,6,4]=>12
[5,2,3,4,1,6]=>6
[5,2,3,4,6,1]=>2
[5,2,3,6,1,4]=>5
[5,2,3,6,4,1]=>9
[5,2,4,1,3,6]=>7
[5,2,4,1,6,3]=>11
[5,2,4,3,1,6]=>14
[5,2,4,3,6,1]=>10
[5,2,4,6,1,3]=>4
[5,2,4,6,3,1]=>8
[5,2,6,1,3,4]=>6
[5,2,6,1,4,3]=>10
[5,2,6,3,1,4]=>13
[5,2,6,3,4,1]=>9
[5,2,6,4,1,3]=>12
[5,2,6,4,3,1]=>16
[5,3,1,2,4,6]=>7
[5,3,1,2,6,4]=>11
[5,3,1,4,2,6]=>14
[5,3,1,4,6,2]=>10
[5,3,1,6,2,4]=>13
[5,3,1,6,4,2]=>17
[5,3,2,1,4,6]=>15
[5,3,2,1,6,4]=>19
[5,3,2,4,1,6]=>13
[5,3,2,4,6,1]=>9
[5,3,2,6,1,4]=>12
[5,3,2,6,4,1]=>16
[5,3,4,1,2,6]=>6
[5,3,4,1,6,2]=>10
[5,3,4,2,1,6]=>13
[5,3,4,2,6,1]=>9
[5,3,4,6,1,2]=>3
[5,3,4,6,2,1]=>7
[5,3,6,1,2,4]=>5
[5,3,6,1,4,2]=>9
[5,3,6,2,1,4]=>12
[5,3,6,2,4,1]=>8
[5,3,6,4,1,2]=>11
[5,3,6,4,2,1]=>15
[5,4,1,2,3,6]=>6
[5,4,1,2,6,3]=>10
[5,4,1,3,2,6]=>13
[5,4,1,3,6,2]=>9
[5,4,1,6,2,3]=>12
[5,4,1,6,3,2]=>16
[5,4,2,1,3,6]=>14
[5,4,2,1,6,3]=>18
[5,4,2,3,1,6]=>12
[5,4,2,3,6,1]=>8
[5,4,2,6,1,3]=>11
[5,4,2,6,3,1]=>15
[5,4,3,1,2,6]=>13
[5,4,3,1,6,2]=>17
[5,4,3,2,1,6]=>20
[5,4,3,2,6,1]=>16
[5,4,3,6,1,2]=>10
[5,4,3,6,2,1]=>14
[5,4,6,1,2,3]=>4
[5,4,6,1,3,2]=>8
[5,4,6,2,1,3]=>11
[5,4,6,2,3,1]=>7
[5,4,6,3,1,2]=>10
[5,4,6,3,2,1]=>14
[5,6,1,2,3,4]=>0
[5,6,1,2,4,3]=>4
[5,6,1,3,2,4]=>7
[5,6,1,3,4,2]=>3
[5,6,1,4,2,3]=>6
[5,6,1,4,3,2]=>10
[5,6,2,1,3,4]=>8
[5,6,2,1,4,3]=>12
[5,6,2,3,1,4]=>6
[5,6,2,3,4,1]=>2
[5,6,2,4,1,3]=>5
[5,6,2,4,3,1]=>9
[5,6,3,1,2,4]=>7
[5,6,3,1,4,2]=>11
[5,6,3,2,1,4]=>14
[5,6,3,2,4,1]=>10
[5,6,3,4,1,2]=>4
[5,6,3,4,2,1]=>8
[5,6,4,1,2,3]=>6
[5,6,4,1,3,2]=>10
[5,6,4,2,1,3]=>13
[5,6,4,2,3,1]=>9
[5,6,4,3,1,2]=>12
[5,6,4,3,2,1]=>16
[6,1,2,3,4,5]=>0
[6,1,2,3,5,4]=>4
[6,1,2,4,3,5]=>7
[6,1,2,4,5,3]=>3
[6,1,2,5,3,4]=>6
[6,1,2,5,4,3]=>10
[6,1,3,2,4,5]=>8
[6,1,3,2,5,4]=>12
[6,1,3,4,2,5]=>6
[6,1,3,4,5,2]=>2
[6,1,3,5,2,4]=>5
[6,1,3,5,4,2]=>9
[6,1,4,2,3,5]=>7
[6,1,4,2,5,3]=>11
[6,1,4,3,2,5]=>14
[6,1,4,3,5,2]=>10
[6,1,4,5,2,3]=>4
[6,1,4,5,3,2]=>8
[6,1,5,2,3,4]=>6
[6,1,5,2,4,3]=>10
[6,1,5,3,2,4]=>13
[6,1,5,3,4,2]=>9
[6,1,5,4,2,3]=>12
[6,1,5,4,3,2]=>16
[6,2,1,3,4,5]=>7
[6,2,1,3,5,4]=>11
[6,2,1,4,3,5]=>14
[6,2,1,4,5,3]=>10
[6,2,1,5,3,4]=>13
[6,2,1,5,4,3]=>17
[6,2,3,1,4,5]=>7
[6,2,3,1,5,4]=>11
[6,2,3,4,1,5]=>5
[6,2,3,4,5,1]=>1
[6,2,3,5,1,4]=>4
[6,2,3,5,4,1]=>8
[6,2,4,1,3,5]=>6
[6,2,4,1,5,3]=>10
[6,2,4,3,1,5]=>13
[6,2,4,3,5,1]=>9
[6,2,4,5,1,3]=>3
[6,2,4,5,3,1]=>7
[6,2,5,1,3,4]=>5
[6,2,5,1,4,3]=>9
[6,2,5,3,1,4]=>12
[6,2,5,3,4,1]=>8
[6,2,5,4,1,3]=>11
[6,2,5,4,3,1]=>15
[6,3,1,2,4,5]=>6
[6,3,1,2,5,4]=>10
[6,3,1,4,2,5]=>13
[6,3,1,4,5,2]=>9
[6,3,1,5,2,4]=>12
[6,3,1,5,4,2]=>16
[6,3,2,1,4,5]=>14
[6,3,2,1,5,4]=>18
[6,3,2,4,1,5]=>12
[6,3,2,4,5,1]=>8
[6,3,2,5,1,4]=>11
[6,3,2,5,4,1]=>15
[6,3,4,1,2,5]=>5
[6,3,4,1,5,2]=>9
[6,3,4,2,1,5]=>12
[6,3,4,2,5,1]=>8
[6,3,4,5,1,2]=>2
[6,3,4,5,2,1]=>6
[6,3,5,1,2,4]=>4
[6,3,5,1,4,2]=>8
[6,3,5,2,1,4]=>11
[6,3,5,2,4,1]=>7
[6,3,5,4,1,2]=>10
[6,3,5,4,2,1]=>14
[6,4,1,2,3,5]=>5
[6,4,1,2,5,3]=>9
[6,4,1,3,2,5]=>12
[6,4,1,3,5,2]=>8
[6,4,1,5,2,3]=>11
[6,4,1,5,3,2]=>15
[6,4,2,1,3,5]=>13
[6,4,2,1,5,3]=>17
[6,4,2,3,1,5]=>11
[6,4,2,3,5,1]=>7
[6,4,2,5,1,3]=>10
[6,4,2,5,3,1]=>14
[6,4,3,1,2,5]=>12
[6,4,3,1,5,2]=>16
[6,4,3,2,1,5]=>19
[6,4,3,2,5,1]=>15
[6,4,3,5,1,2]=>9
[6,4,3,5,2,1]=>13
[6,4,5,1,2,3]=>3
[6,4,5,1,3,2]=>7
[6,4,5,2,1,3]=>10
[6,4,5,2,3,1]=>6
[6,4,5,3,1,2]=>9
[6,4,5,3,2,1]=>13
[6,5,1,2,3,4]=>4
[6,5,1,2,4,3]=>8
[6,5,1,3,2,4]=>11
[6,5,1,3,4,2]=>7
[6,5,1,4,2,3]=>10
[6,5,1,4,3,2]=>14
[6,5,2,1,3,4]=>12
[6,5,2,1,4,3]=>16
[6,5,2,3,1,4]=>10
[6,5,2,3,4,1]=>6
[6,5,2,4,1,3]=>9
[6,5,2,4,3,1]=>13
[6,5,3,1,2,4]=>11
[6,5,3,1,4,2]=>15
[6,5,3,2,1,4]=>18
[6,5,3,2,4,1]=>14
[6,5,3,4,1,2]=>8
[6,5,3,4,2,1]=>12
[6,5,4,1,2,3]=>10
[6,5,4,1,3,2]=>14
[6,5,4,2,1,3]=>17
[6,5,4,2,3,1]=>13
[6,5,4,3,1,2]=>16
[6,5,4,3,2,1]=>20
[1,2,3,4,5,6,7]=>0
[1,2,3,4,5,7,6]=>5
[1,2,3,4,6,5,7]=>9
[1,2,3,4,6,7,5]=>4
[1,2,3,4,7,5,6]=>8
[1,2,3,4,7,6,5]=>13
[1,2,3,5,4,6,7]=>11
[1,2,3,5,4,7,6]=>16
[1,2,3,5,6,4,7]=>8
[1,2,3,5,6,7,4]=>3
[1,2,3,5,7,4,6]=>7
[1,2,3,5,7,6,4]=>12
[1,2,3,6,4,5,7]=>10
[1,2,3,6,4,7,5]=>15
[1,2,3,6,5,4,7]=>19
[1,2,3,6,5,7,4]=>14
[1,2,3,6,7,4,5]=>6
[1,2,3,6,7,5,4]=>11
[1,2,3,7,4,5,6]=>9
[1,2,3,7,4,6,5]=>14
[1,2,3,7,5,4,6]=>18
[1,2,3,7,5,6,4]=>13
[1,2,3,7,6,4,5]=>17
[1,2,3,7,6,5,4]=>22
[1,2,4,3,5,6,7]=>11
[1,2,4,3,5,7,6]=>16
[1,2,4,3,6,5,7]=>20
[1,2,4,3,6,7,5]=>15
[1,2,4,3,7,5,6]=>19
[1,2,4,3,7,6,5]=>24
[1,2,4,5,3,6,7]=>10
[1,2,4,5,3,7,6]=>15
[1,2,4,5,6,3,7]=>7
[1,2,4,5,6,7,3]=>2
[1,2,4,5,7,3,6]=>6
[1,2,4,5,7,6,3]=>11
[1,2,4,6,3,5,7]=>9
[1,2,4,6,3,7,5]=>14
[1,2,4,6,5,3,7]=>18
[1,2,4,6,5,7,3]=>13
[1,2,4,6,7,3,5]=>5
[1,2,4,6,7,5,3]=>10
[1,2,4,7,3,5,6]=>8
[1,2,4,7,3,6,5]=>13
[1,2,4,7,5,3,6]=>17
[1,2,4,7,5,6,3]=>12
[1,2,4,7,6,3,5]=>16
[1,2,4,7,6,5,3]=>21
[1,2,5,3,4,6,7]=>10
[1,2,5,3,4,7,6]=>15
[1,2,5,3,6,4,7]=>19
[1,2,5,3,6,7,4]=>14
[1,2,5,3,7,4,6]=>18
[1,2,5,3,7,6,4]=>23
[1,2,5,4,3,6,7]=>21
[1,2,5,4,3,7,6]=>26
[1,2,5,4,6,3,7]=>18
[1,2,5,4,6,7,3]=>13
[1,2,5,4,7,3,6]=>17
[1,2,5,4,7,6,3]=>22
[1,2,5,6,3,4,7]=>8
[1,2,5,6,3,7,4]=>13
[1,2,5,6,4,3,7]=>17
[1,2,5,6,4,7,3]=>12
[1,2,5,6,7,3,4]=>4
[1,2,5,6,7,4,3]=>9
[1,2,5,7,3,4,6]=>7
[1,2,5,7,3,6,4]=>12
[1,2,5,7,4,3,6]=>16
[1,2,5,7,4,6,3]=>11
[1,2,5,7,6,3,4]=>15
[1,2,5,7,6,4,3]=>20
[1,2,6,3,4,5,7]=>9
[1,2,6,3,4,7,5]=>14
[1,2,6,3,5,4,7]=>18
[1,2,6,3,5,7,4]=>13
[1,2,6,3,7,4,5]=>17
[1,2,6,3,7,5,4]=>22
[1,2,6,4,3,5,7]=>20
[1,2,6,4,3,7,5]=>25
[1,2,6,4,5,3,7]=>17
[1,2,6,4,5,7,3]=>12
[1,2,6,4,7,3,5]=>16
[1,2,6,4,7,5,3]=>21
[1,2,6,5,3,4,7]=>19
[1,2,6,5,3,7,4]=>24
[1,2,6,5,4,3,7]=>28
[1,2,6,5,4,7,3]=>23
[1,2,6,5,7,3,4]=>15
[1,2,6,5,7,4,3]=>20
[1,2,6,7,3,4,5]=>6
[1,2,6,7,3,5,4]=>11
[1,2,6,7,4,3,5]=>15
[1,2,6,7,4,5,3]=>10
[1,2,6,7,5,3,4]=>14
[1,2,6,7,5,4,3]=>19
[1,2,7,3,4,5,6]=>8
[1,2,7,3,4,6,5]=>13
[1,2,7,3,5,4,6]=>17
[1,2,7,3,5,6,4]=>12
[1,2,7,3,6,4,5]=>16
[1,2,7,3,6,5,4]=>21
[1,2,7,4,3,5,6]=>19
[1,2,7,4,3,6,5]=>24
[1,2,7,4,5,3,6]=>16
[1,2,7,4,5,6,3]=>11
[1,2,7,4,6,3,5]=>15
[1,2,7,4,6,5,3]=>20
[1,2,7,5,3,4,6]=>18
[1,2,7,5,3,6,4]=>23
[1,2,7,5,4,3,6]=>27
[1,2,7,5,4,6,3]=>22
[1,2,7,5,6,3,4]=>14
[1,2,7,5,6,4,3]=>19
[1,2,7,6,3,4,5]=>17
[1,2,7,6,3,5,4]=>22
[1,2,7,6,4,3,5]=>26
[1,2,7,6,4,5,3]=>21
[1,2,7,6,5,3,4]=>25
[1,2,7,6,5,4,3]=>30
[1,3,2,4,5,6,7]=>9
[1,3,2,4,5,7,6]=>14
[1,3,2,4,6,5,7]=>18
[1,3,2,4,6,7,5]=>13
[1,3,2,4,7,5,6]=>17
[1,3,2,4,7,6,5]=>22
[1,3,2,5,4,6,7]=>20
[1,3,2,5,4,7,6]=>25
[1,3,2,5,6,4,7]=>17
[1,3,2,5,6,7,4]=>12
[1,3,2,5,7,4,6]=>16
[1,3,2,5,7,6,4]=>21
[1,3,2,6,4,5,7]=>19
[1,3,2,6,4,7,5]=>24
[1,3,2,6,5,4,7]=>28
[1,3,2,6,5,7,4]=>23
[1,3,2,6,7,4,5]=>15
[1,3,2,6,7,5,4]=>20
[1,3,2,7,4,5,6]=>18
[1,3,2,7,4,6,5]=>23
[1,3,2,7,5,4,6]=>27
[1,3,2,7,5,6,4]=>22
[1,3,2,7,6,4,5]=>26
[1,3,2,7,6,5,4]=>31
[1,3,4,2,5,6,7]=>10
[1,3,4,2,5,7,6]=>15
[1,3,4,2,6,5,7]=>19
[1,3,4,2,6,7,5]=>14
[1,3,4,2,7,5,6]=>18
[1,3,4,2,7,6,5]=>23
[1,3,4,5,2,6,7]=>9
[1,3,4,5,2,7,6]=>14
[1,3,4,5,6,2,7]=>6
[1,3,4,5,6,7,2]=>1
[1,3,4,5,7,2,6]=>5
[1,3,4,5,7,6,2]=>10
[1,3,4,6,2,5,7]=>8
[1,3,4,6,2,7,5]=>13
[1,3,4,6,5,2,7]=>17
[1,3,4,6,5,7,2]=>12
[1,3,4,6,7,2,5]=>4
[1,3,4,6,7,5,2]=>9
[1,3,4,7,2,5,6]=>7
[1,3,4,7,2,6,5]=>12
[1,3,4,7,5,2,6]=>16
[1,3,4,7,5,6,2]=>11
[1,3,4,7,6,2,5]=>15
[1,3,4,7,6,5,2]=>20
[1,3,5,2,4,6,7]=>9
[1,3,5,2,4,7,6]=>14
[1,3,5,2,6,4,7]=>18
[1,3,5,2,6,7,4]=>13
[1,3,5,2,7,4,6]=>17
[1,3,5,2,7,6,4]=>22
[1,3,5,4,2,6,7]=>20
[1,3,5,4,2,7,6]=>25
[1,3,5,4,6,2,7]=>17
[1,3,5,4,6,7,2]=>12
[1,3,5,4,7,2,6]=>16
[1,3,5,4,7,6,2]=>21
[1,3,5,6,2,4,7]=>7
[1,3,5,6,2,7,4]=>12
[1,3,5,6,4,2,7]=>16
[1,3,5,6,4,7,2]=>11
[1,3,5,6,7,2,4]=>3
[1,3,5,6,7,4,2]=>8
[1,3,5,7,2,4,6]=>6
[1,3,5,7,2,6,4]=>11
[1,3,5,7,4,2,6]=>15
[1,3,5,7,4,6,2]=>10
[1,3,5,7,6,2,4]=>14
[1,3,5,7,6,4,2]=>19
[1,3,6,2,4,5,7]=>8
[1,3,6,2,4,7,5]=>13
[1,3,6,2,5,4,7]=>17
[1,3,6,2,5,7,4]=>12
[1,3,6,2,7,4,5]=>16
[1,3,6,2,7,5,4]=>21
[1,3,6,4,2,5,7]=>19
[1,3,6,4,2,7,5]=>24
[1,3,6,4,5,2,7]=>16
[1,3,6,4,5,7,2]=>11
[1,3,6,4,7,2,5]=>15
[1,3,6,4,7,5,2]=>20
[1,3,6,5,2,4,7]=>18
[1,3,6,5,2,7,4]=>23
[1,3,6,5,4,2,7]=>27
[1,3,6,5,4,7,2]=>22
[1,3,6,5,7,2,4]=>14
[1,3,6,5,7,4,2]=>19
[1,3,6,7,2,4,5]=>5
[1,3,6,7,2,5,4]=>10
[1,3,6,7,4,2,5]=>14
[1,3,6,7,4,5,2]=>9
[1,3,6,7,5,2,4]=>13
[1,3,6,7,5,4,2]=>18
[1,3,7,2,4,5,6]=>7
[1,3,7,2,4,6,5]=>12
[1,3,7,2,5,4,6]=>16
[1,3,7,2,5,6,4]=>11
[1,3,7,2,6,4,5]=>15
[1,3,7,2,6,5,4]=>20
[1,3,7,4,2,5,6]=>18
[1,3,7,4,2,6,5]=>23
[1,3,7,4,5,2,6]=>15
[1,3,7,4,5,6,2]=>10
[1,3,7,4,6,2,5]=>14
[1,3,7,4,6,5,2]=>19
[1,3,7,5,2,4,6]=>17
[1,3,7,5,2,6,4]=>22
[1,3,7,5,4,2,6]=>26
[1,3,7,5,4,6,2]=>21
[1,3,7,5,6,2,4]=>13
[1,3,7,5,6,4,2]=>18
[1,3,7,6,2,4,5]=>16
[1,3,7,6,2,5,4]=>21
[1,3,7,6,4,2,5]=>25
[1,3,7,6,4,5,2]=>20
[1,3,7,6,5,2,4]=>24
[1,3,7,6,5,4,2]=>29
[1,4,2,3,5,6,7]=>8
[1,4,2,3,5,7,6]=>13
[1,4,2,3,6,5,7]=>17
[1,4,2,3,6,7,5]=>12
[1,4,2,3,7,5,6]=>16
[1,4,2,3,7,6,5]=>21
[1,4,2,5,3,6,7]=>19
[1,4,2,5,3,7,6]=>24
[1,4,2,5,6,3,7]=>16
[1,4,2,5,6,7,3]=>11
[1,4,2,5,7,3,6]=>15
[1,4,2,5,7,6,3]=>20
[1,4,2,6,3,5,7]=>18
[1,4,2,6,3,7,5]=>23
[1,4,2,6,5,3,7]=>27
[1,4,2,6,5,7,3]=>22
[1,4,2,6,7,3,5]=>14
[1,4,2,6,7,5,3]=>19
[1,4,2,7,3,5,6]=>17
[1,4,2,7,3,6,5]=>22
[1,4,2,7,5,3,6]=>26
[1,4,2,7,5,6,3]=>21
[1,4,2,7,6,3,5]=>25
[1,4,2,7,6,5,3]=>30
[1,4,3,2,5,6,7]=>19
[1,4,3,2,5,7,6]=>24
[1,4,3,2,6,5,7]=>28
[1,4,3,2,6,7,5]=>23
[1,4,3,2,7,5,6]=>27
[1,4,3,2,7,6,5]=>32
[1,4,3,5,2,6,7]=>18
[1,4,3,5,2,7,6]=>23
[1,4,3,5,6,2,7]=>15
[1,4,3,5,6,7,2]=>10
[1,4,3,5,7,2,6]=>14
[1,4,3,5,7,6,2]=>19
[1,4,3,6,2,5,7]=>17
[1,4,3,6,2,7,5]=>22
[1,4,3,6,5,2,7]=>26
[1,4,3,6,5,7,2]=>21
[1,4,3,6,7,2,5]=>13
[1,4,3,6,7,5,2]=>18
[1,4,3,7,2,5,6]=>16
[1,4,3,7,2,6,5]=>21
[1,4,3,7,5,2,6]=>25
[1,4,3,7,5,6,2]=>20
[1,4,3,7,6,2,5]=>24
[1,4,3,7,6,5,2]=>29
[1,4,5,2,3,6,7]=>8
[1,4,5,2,3,7,6]=>13
[1,4,5,2,6,3,7]=>17
[1,4,5,2,6,7,3]=>12
[1,4,5,2,7,3,6]=>16
[1,4,5,2,7,6,3]=>21
[1,4,5,3,2,6,7]=>19
[1,4,5,3,2,7,6]=>24
[1,4,5,3,6,2,7]=>16
[1,4,5,3,6,7,2]=>11
[1,4,5,3,7,2,6]=>15
[1,4,5,3,7,6,2]=>20
[1,4,5,6,2,3,7]=>6
[1,4,5,6,2,7,3]=>11
[1,4,5,6,3,2,7]=>15
[1,4,5,6,3,7,2]=>10
[1,4,5,6,7,2,3]=>2
[1,4,5,6,7,3,2]=>7
[1,4,5,7,2,3,6]=>5
[1,4,5,7,2,6,3]=>10
[1,4,5,7,3,2,6]=>14
[1,4,5,7,3,6,2]=>9
[1,4,5,7,6,2,3]=>13
[1,4,5,7,6,3,2]=>18
[1,4,6,2,3,5,7]=>7
[1,4,6,2,3,7,5]=>12
[1,4,6,2,5,3,7]=>16
[1,4,6,2,5,7,3]=>11
[1,4,6,2,7,3,5]=>15
[1,4,6,2,7,5,3]=>20
[1,4,6,3,2,5,7]=>18
[1,4,6,3,2,7,5]=>23
[1,4,6,3,5,2,7]=>15
[1,4,6,3,5,7,2]=>10
[1,4,6,3,7,2,5]=>14
[1,4,6,3,7,5,2]=>19
[1,4,6,5,2,3,7]=>17
[1,4,6,5,2,7,3]=>22
[1,4,6,5,3,2,7]=>26
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
A descent variant minus the number of inversions.
This statistic is defined for general finite crystallographic root system $\Phi$ with Weyl group $W$ as follows: Let $2\rho = \sum_{\beta \in \Phi^+} \beta = \sum_{\alpha\in\Delta}b_\alpha \alpha$ be the sum of the positive roots expressed in the simple roots.
For $w \in W$ this statistic is then
$$\operatorname{stat}(w) = \sum_{\alpha\in\Delta\,:\,w(\alpha) \in \Phi^-}b_\alpha - \ell(w)\,,$$
where the sum ranges over all descents of $w$ and $\ell(w)$ is the Coxeter length.
It was shown in [1], that for irreducible groups, it holds that
$$\sum_{w\in W} q^{\operatorname{stat}(w)} = f\prod_{\alpha \in \Delta} \frac{1-q^{b_\alpha}}{1-q^{e_\alpha}}\,,$$
where $\{ e_\alpha \mid \alpha \in \Delta\}$ are the exponents of the group and $f$ is its index of connection, i.e., the index of the root lattice inside the weight lattice.
For a permutation $\sigma \in S_n$, this becomes
$$\operatorname{stat}(\sigma) = \sum_{i \in \operatorname{Des}(\sigma)}i\cdot(n-i) - \operatorname{inv}(\sigma)\,.$$
This statistic is defined for general finite crystallographic root system $\Phi$ with Weyl group $W$ as follows: Let $2\rho = \sum_{\beta \in \Phi^+} \beta = \sum_{\alpha\in\Delta}b_\alpha \alpha$ be the sum of the positive roots expressed in the simple roots.
For $w \in W$ this statistic is then
$$\operatorname{stat}(w) = \sum_{\alpha\in\Delta\,:\,w(\alpha) \in \Phi^-}b_\alpha - \ell(w)\,,$$
where the sum ranges over all descents of $w$ and $\ell(w)$ is the Coxeter length.
It was shown in [1], that for irreducible groups, it holds that
$$\sum_{w\in W} q^{\operatorname{stat}(w)} = f\prod_{\alpha \in \Delta} \frac{1-q^{b_\alpha}}{1-q^{e_\alpha}}\,,$$
where $\{ e_\alpha \mid \alpha \in \Delta\}$ are the exponents of the group and $f$ is its index of connection, i.e., the index of the root lattice inside the weight lattice.
For a permutation $\sigma \in S_n$, this becomes
$$\operatorname{stat}(\sigma) = \sum_{i \in \operatorname{Des}(\sigma)}i\cdot(n-i) - \operatorname{inv}(\sigma)\,.$$
References
[1] Stembridge, J. R., Waugh, D. J. A Weyl group generating function that ought to be better known MathSciNet:1692145
Code
def statistic(pi):
n = len(pi)
return sum(i*(n-i) for i in pi.descents()) - pi.number_of_inversions()
Created
Aug 08, 2023 at 16:30 by Christian Stump
Updated
Aug 08, 2023 at 16:30 by Christian Stump
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!