Identifier
- St001412: Permutations ⟶ ℤ
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]=>6
[3,4,2,1]=>7
[4,1,2,3]=>6
[4,1,3,2]=>7
[4,2,1,3]=>7
[4,2,3,1]=>9
[4,3,1,2]=>7
[4,3,2,1]=>8
[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]=>6
[1,4,5,3,2]=>7
[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]=>7
[1,5,4,3,2]=>8
[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]=>9
[2,4,5,3,1]=>11
[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]=>10
[2,5,4,3,1]=>12
[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]=>6
[3,4,1,5,2]=>9
[3,4,2,1,5]=>7
[3,4,2,5,1]=>11
[3,4,5,1,2]=>9
[3,4,5,2,1]=>10
[3,5,1,2,4]=>9
[3,5,1,4,2]=>11
[3,5,2,1,4]=>10
[3,5,2,4,1]=>13
[3,5,4,1,2]=>10
[3,5,4,2,1]=>11
[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]=>9
[4,1,5,3,2]=>10
[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]=>11
[4,2,5,3,1]=>13
[4,3,1,2,5]=>7
[4,3,1,5,2]=>10
[4,3,2,1,5]=>8
[4,3,2,5,1]=>12
[4,3,5,1,2]=>10
[4,3,5,2,1]=>11
[4,5,1,2,3]=>9
[4,5,1,3,2]=>10
[4,5,2,1,3]=>10
[4,5,2,3,1]=>12
[4,5,3,1,2]=>10
[4,5,3,2,1]=>11
[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]=>11
[5,1,4,3,2]=>12
[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]=>13
[5,2,4,3,1]=>15
[5,3,1,2,4]=>11
[5,3,1,4,2]=>13
[5,3,2,1,4]=>12
[5,3,2,4,1]=>15
[5,3,4,1,2]=>12
[5,3,4,2,1]=>13
[5,4,1,2,3]=>10
[5,4,1,3,2]=>11
[5,4,2,1,3]=>11
[5,4,2,3,1]=>13
[5,4,3,1,2]=>11
[5,4,3,2,1]=>12
[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]=>6
[1,2,5,6,4,3]=>7
[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]=>7
[1,2,6,5,4,3]=>8
[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]=>9
[1,3,5,6,4,2]=>11
[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]=>10
[1,3,6,5,4,2]=>12
[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]=>6
[1,4,5,2,6,3]=>9
[1,4,5,3,2,6]=>7
[1,4,5,3,6,2]=>11
[1,4,5,6,2,3]=>9
[1,4,5,6,3,2]=>10
[1,4,6,2,3,5]=>9
[1,4,6,2,5,3]=>11
[1,4,6,3,2,5]=>10
[1,4,6,3,5,2]=>13
[1,4,6,5,2,3]=>10
[1,4,6,5,3,2]=>11
[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]=>9
[1,5,2,6,4,3]=>10
[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]=>11
[1,5,3,6,4,2]=>13
[1,5,4,2,3,6]=>7
[1,5,4,2,6,3]=>10
[1,5,4,3,2,6]=>8
[1,5,4,3,6,2]=>12
[1,5,4,6,2,3]=>10
[1,5,4,6,3,2]=>11
[1,5,6,2,3,4]=>9
[1,5,6,2,4,3]=>10
[1,5,6,3,2,4]=>10
[1,5,6,3,4,2]=>12
[1,5,6,4,2,3]=>10
[1,5,6,4,3,2]=>11
[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]=>11
[1,6,2,5,4,3]=>12
[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]=>13
[1,6,3,5,4,2]=>15
[1,6,4,2,3,5]=>11
[1,6,4,2,5,3]=>13
[1,6,4,3,2,5]=>12
[1,6,4,3,5,2]=>15
[1,6,4,5,2,3]=>12
[1,6,4,5,3,2]=>13
[1,6,5,2,3,4]=>10
[1,6,5,2,4,3]=>11
[1,6,5,3,2,4]=>11
[1,6,5,3,4,2]=>13
[1,6,5,4,2,3]=>11
[1,6,5,4,3,2]=>12
[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]=>7
[2,1,5,6,4,3]=>8
[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]=>8
[2,1,6,5,4,3]=>9
[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]=>13
[2,3,5,6,4,1]=>16
[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]=>14
[2,3,6,5,4,1]=>17
[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]=>9
[2,4,5,1,6,3]=>12
[2,4,5,3,1,6]=>11
[2,4,5,3,6,1]=>16
[2,4,5,6,1,3]=>13
[2,4,5,6,3,1]=>15
[2,4,6,1,3,5]=>12
[2,4,6,1,5,3]=>14
[2,4,6,3,1,5]=>14
[2,4,6,3,5,1]=>18
[2,4,6,5,1,3]=>14
[2,4,6,5,3,1]=>16
[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]=>11
[2,5,1,6,4,3]=>12
[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]=>15
[2,5,3,6,4,1]=>18
[2,5,4,1,3,6]=>10
[2,5,4,1,6,3]=>13
[2,5,4,3,1,6]=>12
[2,5,4,3,6,1]=>17
[2,5,4,6,1,3]=>14
[2,5,4,6,3,1]=>16
[2,5,6,1,3,4]=>12
[2,5,6,1,4,3]=>13
[2,5,6,3,1,4]=>14
[2,5,6,3,4,1]=>17
[2,5,6,4,1,3]=>14
[2,5,6,4,3,1]=>16
[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]=>13
[2,6,1,5,4,3]=>14
[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]=>17
[2,6,3,5,4,1]=>20
[2,6,4,1,3,5]=>14
[2,6,4,1,5,3]=>16
[2,6,4,3,1,5]=>16
[2,6,4,3,5,1]=>20
[2,6,4,5,1,3]=>16
[2,6,4,5,3,1]=>18
[2,6,5,1,3,4]=>13
[2,6,5,1,4,3]=>14
[2,6,5,3,1,4]=>15
[2,6,5,3,4,1]=>18
[2,6,5,4,1,3]=>15
[2,6,5,4,3,1]=>17
[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]=>11
[3,1,5,6,4,2]=>13
[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]=>12
[3,1,6,5,4,2]=>14
[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]=>14
[3,2,5,6,4,1]=>17
[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]=>15
[3,2,6,5,4,1]=>18
[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]=>13
[3,4,1,6,2,5]=>11
[3,4,1,6,5,2]=>14
[3,4,2,1,5,6]=>7
[3,4,2,1,6,5]=>8
[3,4,2,5,1,6]=>11
[3,4,2,5,6,1]=>16
[3,4,2,6,1,5]=>13
[3,4,2,6,5,1]=>17
[3,4,5,1,2,6]=>9
[3,4,5,1,6,2]=>13
[3,4,5,2,1,6]=>10
[3,4,5,2,6,1]=>15
[3,4,5,6,1,2]=>12
[3,4,5,6,2,1]=>13
[3,4,6,1,2,5]=>12
[3,4,6,1,5,2]=>15
[3,4,6,2,1,5]=>13
[3,4,6,2,5,1]=>17
[3,4,6,5,1,2]=>13
[3,4,6,5,2,1]=>14
[3,5,1,2,4,6]=>9
[3,5,1,2,6,4]=>11
[3,5,1,4,2,6]=>11
[3,5,1,4,6,2]=>15
[3,5,1,6,2,4]=>13
[3,5,1,6,4,2]=>15
[3,5,2,1,4,6]=>10
[3,5,2,1,6,4]=>12
[3,5,2,4,1,6]=>13
[3,5,2,4,6,1]=>18
[3,5,2,6,1,4]=>15
[3,5,2,6,4,1]=>18
[3,5,4,1,2,6]=>10
[3,5,4,1,6,2]=>14
[3,5,4,2,1,6]=>11
[3,5,4,2,6,1]=>16
[3,5,4,6,1,2]=>13
[3,5,4,6,2,1]=>14
[3,5,6,1,2,4]=>12
[3,5,6,1,4,2]=>14
[3,5,6,2,1,4]=>13
[3,5,6,2,4,1]=>16
[3,5,6,4,1,2]=>13
[3,5,6,4,2,1]=>14
[3,6,1,2,4,5]=>13
[3,6,1,2,5,4]=>14
[3,6,1,4,2,5]=>15
[3,6,1,4,5,2]=>18
[3,6,1,5,2,4]=>15
[3,6,1,5,4,2]=>17
[3,6,2,1,4,5]=>14
[3,6,2,1,5,4]=>15
[3,6,2,4,1,5]=>17
[3,6,2,4,5,1]=>21
[3,6,2,5,1,4]=>17
[3,6,2,5,4,1]=>20
[3,6,4,1,2,5]=>14
[3,6,4,1,5,2]=>17
[3,6,4,2,1,5]=>15
[3,6,4,2,5,1]=>19
[3,6,4,5,1,2]=>15
[3,6,4,5,2,1]=>16
[3,6,5,1,2,4]=>13
[3,6,5,1,4,2]=>15
[3,6,5,2,1,4]=>14
[3,6,5,2,4,1]=>17
[3,6,5,4,1,2]=>14
[3,6,5,4,2,1]=>15
[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]=>9
[4,1,5,2,6,3]=>12
[4,1,5,3,2,6]=>10
[4,1,5,3,6,2]=>14
[4,1,5,6,2,3]=>12
[4,1,5,6,3,2]=>13
[4,1,6,2,3,5]=>12
[4,1,6,2,5,3]=>14
[4,1,6,3,2,5]=>13
[4,1,6,3,5,2]=>16
[4,1,6,5,2,3]=>13
[4,1,6,5,3,2]=>14
[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]=>11
[4,2,5,1,6,3]=>14
[4,2,5,3,1,6]=>13
[4,2,5,3,6,1]=>18
[4,2,5,6,1,3]=>15
[4,2,5,6,3,1]=>17
[4,2,6,1,3,5]=>14
[4,2,6,1,5,3]=>16
[4,2,6,3,1,5]=>16
[4,2,6,3,5,1]=>20
[4,2,6,5,1,3]=>16
[4,2,6,5,3,1]=>18
[4,3,1,2,5,6]=>7
[4,3,1,2,6,5]=>8
[4,3,1,5,2,6]=>10
[4,3,1,5,6,2]=>14
[4,3,1,6,2,5]=>12
[4,3,1,6,5,2]=>15
[4,3,2,1,5,6]=>8
[4,3,2,1,6,5]=>9
[4,3,2,5,1,6]=>12
[4,3,2,5,6,1]=>17
[4,3,2,6,1,5]=>14
[4,3,2,6,5,1]=>18
[4,3,5,1,2,6]=>10
[4,3,5,1,6,2]=>14
[4,3,5,2,1,6]=>11
[4,3,5,2,6,1]=>16
[4,3,5,6,1,2]=>13
[4,3,5,6,2,1]=>14
[4,3,6,1,2,5]=>13
[4,3,6,1,5,2]=>16
[4,3,6,2,1,5]=>14
[4,3,6,2,5,1]=>18
[4,3,6,5,1,2]=>14
[4,3,6,5,2,1]=>15
[4,5,1,2,3,6]=>9
[4,5,1,2,6,3]=>12
[4,5,1,3,2,6]=>10
[4,5,1,3,6,2]=>14
[4,5,1,6,2,3]=>12
[4,5,1,6,3,2]=>13
[4,5,2,1,3,6]=>10
[4,5,2,1,6,3]=>13
[4,5,2,3,1,6]=>12
[4,5,2,3,6,1]=>17
[4,5,2,6,1,3]=>14
[4,5,2,6,3,1]=>16
[4,5,3,1,2,6]=>10
[4,5,3,1,6,2]=>14
[4,5,3,2,1,6]=>11
[4,5,3,2,6,1]=>16
[4,5,3,6,1,2]=>13
[4,5,3,6,2,1]=>14
[4,5,6,1,2,3]=>12
[4,5,6,1,3,2]=>13
[4,5,6,2,1,3]=>13
[4,5,6,2,3,1]=>15
[4,5,6,3,1,2]=>13
[4,5,6,3,2,1]=>14
[4,6,1,2,3,5]=>13
[4,6,1,2,5,3]=>15
[4,6,1,3,2,5]=>14
[4,6,1,3,5,2]=>17
[4,6,1,5,2,3]=>14
[4,6,1,5,3,2]=>15
[4,6,2,1,3,5]=>14
[4,6,2,1,5,3]=>16
[4,6,2,3,1,5]=>16
[4,6,2,3,5,1]=>20
[4,6,2,5,1,3]=>16
[4,6,2,5,3,1]=>18
[4,6,3,1,2,5]=>14
[4,6,3,1,5,2]=>17
[4,6,3,2,1,5]=>15
[4,6,3,2,5,1]=>19
[4,6,3,5,1,2]=>15
[4,6,3,5,2,1]=>16
[4,6,5,1,2,3]=>13
[4,6,5,1,3,2]=>14
[4,6,5,2,1,3]=>14
[4,6,5,2,3,1]=>16
[4,6,5,3,1,2]=>14
[4,6,5,3,2,1]=>15
[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]=>13
[5,1,2,6,4,3]=>14
[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]=>15
[5,1,3,6,4,2]=>17
[5,1,4,2,3,6]=>11
[5,1,4,2,6,3]=>14
[5,1,4,3,2,6]=>12
[5,1,4,3,6,2]=>16
[5,1,4,6,2,3]=>14
[5,1,4,6,3,2]=>15
[5,1,6,2,3,4]=>13
[5,1,6,2,4,3]=>14
[5,1,6,3,2,4]=>14
[5,1,6,3,4,2]=>16
[5,1,6,4,2,3]=>14
[5,1,6,4,3,2]=>15
[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]=>14
[5,2,1,6,4,3]=>15
[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]=>18
[5,2,3,6,4,1]=>21
[5,2,4,1,3,6]=>13
[5,2,4,1,6,3]=>16
[5,2,4,3,1,6]=>15
[5,2,4,3,6,1]=>20
[5,2,4,6,1,3]=>17
[5,2,4,6,3,1]=>19
[5,2,6,1,3,4]=>15
[5,2,6,1,4,3]=>16
[5,2,6,3,1,4]=>17
[5,2,6,3,4,1]=>20
[5,2,6,4,1,3]=>17
[5,2,6,4,3,1]=>19
[5,3,1,2,4,6]=>11
[5,3,1,2,6,4]=>13
[5,3,1,4,2,6]=>13
[5,3,1,4,6,2]=>17
[5,3,1,6,2,4]=>15
[5,3,1,6,4,2]=>17
[5,3,2,1,4,6]=>12
[5,3,2,1,6,4]=>14
[5,3,2,4,1,6]=>15
[5,3,2,4,6,1]=>20
[5,3,2,6,1,4]=>17
[5,3,2,6,4,1]=>20
[5,3,4,1,2,6]=>12
[5,3,4,1,6,2]=>16
[5,3,4,2,1,6]=>13
[5,3,4,2,6,1]=>18
[5,3,4,6,1,2]=>15
[5,3,4,6,2,1]=>16
[5,3,6,1,2,4]=>14
[5,3,6,1,4,2]=>16
[5,3,6,2,1,4]=>15
[5,3,6,2,4,1]=>18
[5,3,6,4,1,2]=>15
[5,3,6,4,2,1]=>16
[5,4,1,2,3,6]=>10
[5,4,1,2,6,3]=>13
[5,4,1,3,2,6]=>11
[5,4,1,3,6,2]=>15
[5,4,1,6,2,3]=>13
[5,4,1,6,3,2]=>14
[5,4,2,1,3,6]=>11
[5,4,2,1,6,3]=>14
[5,4,2,3,1,6]=>13
[5,4,2,3,6,1]=>18
[5,4,2,6,1,3]=>15
[5,4,2,6,3,1]=>17
[5,4,3,1,2,6]=>11
[5,4,3,1,6,2]=>15
[5,4,3,2,1,6]=>12
[5,4,3,2,6,1]=>17
[5,4,3,6,1,2]=>14
[5,4,3,6,2,1]=>15
[5,4,6,1,2,3]=>13
[5,4,6,1,3,2]=>14
[5,4,6,2,1,3]=>14
[5,4,6,2,3,1]=>16
[5,4,6,3,1,2]=>14
[5,4,6,3,2,1]=>15
[5,6,1,2,3,4]=>12
[5,6,1,2,4,3]=>13
[5,6,1,3,2,4]=>13
[5,6,1,3,4,2]=>15
[5,6,1,4,2,3]=>13
[5,6,1,4,3,2]=>14
[5,6,2,1,3,4]=>13
[5,6,2,1,4,3]=>14
[5,6,2,3,1,4]=>15
[5,6,2,3,4,1]=>18
[5,6,2,4,1,3]=>15
[5,6,2,4,3,1]=>17
[5,6,3,1,2,4]=>13
[5,6,3,1,4,2]=>15
[5,6,3,2,1,4]=>14
[5,6,3,2,4,1]=>17
[5,6,3,4,1,2]=>14
[5,6,3,4,2,1]=>15
[5,6,4,1,2,3]=>13
[5,6,4,1,3,2]=>14
[5,6,4,2,1,3]=>14
[5,6,4,2,3,1]=>16
[5,6,4,3,1,2]=>14
[5,6,4,3,2,1]=>15
[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]=>16
[6,1,2,5,4,3]=>17
[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]=>18
[6,1,3,5,4,2]=>20
[6,1,4,2,3,5]=>16
[6,1,4,2,5,3]=>18
[6,1,4,3,2,5]=>17
[6,1,4,3,5,2]=>20
[6,1,4,5,2,3]=>17
[6,1,4,5,3,2]=>18
[6,1,5,2,3,4]=>15
[6,1,5,2,4,3]=>16
[6,1,5,3,2,4]=>16
[6,1,5,3,4,2]=>18
[6,1,5,4,2,3]=>16
[6,1,5,4,3,2]=>17
[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]=>17
[6,2,1,5,4,3]=>18
[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]=>21
[6,2,3,5,4,1]=>24
[6,2,4,1,3,5]=>18
[6,2,4,1,5,3]=>20
[6,2,4,3,1,5]=>20
[6,2,4,3,5,1]=>24
[6,2,4,5,1,3]=>20
[6,2,4,5,3,1]=>22
[6,2,5,1,3,4]=>17
[6,2,5,1,4,3]=>18
[6,2,5,3,1,4]=>19
[6,2,5,3,4,1]=>22
[6,2,5,4,1,3]=>19
[6,2,5,4,3,1]=>21
[6,3,1,2,4,5]=>16
[6,3,1,2,5,4]=>17
[6,3,1,4,2,5]=>18
[6,3,1,4,5,2]=>21
[6,3,1,5,2,4]=>18
[6,3,1,5,4,2]=>20
[6,3,2,1,4,5]=>17
[6,3,2,1,5,4]=>18
[6,3,2,4,1,5]=>20
[6,3,2,4,5,1]=>24
[6,3,2,5,1,4]=>20
[6,3,2,5,4,1]=>23
[6,3,4,1,2,5]=>17
[6,3,4,1,5,2]=>20
[6,3,4,2,1,5]=>18
[6,3,4,2,5,1]=>22
[6,3,4,5,1,2]=>18
[6,3,4,5,2,1]=>19
[6,3,5,1,2,4]=>16
[6,3,5,1,4,2]=>18
[6,3,5,2,1,4]=>17
[6,3,5,2,4,1]=>20
[6,3,5,4,1,2]=>17
[6,3,5,4,2,1]=>18
[6,4,1,2,3,5]=>15
[6,4,1,2,5,3]=>17
[6,4,1,3,2,5]=>16
[6,4,1,3,5,2]=>19
[6,4,1,5,2,3]=>16
[6,4,1,5,3,2]=>17
[6,4,2,1,3,5]=>16
[6,4,2,1,5,3]=>18
[6,4,2,3,1,5]=>18
[6,4,2,3,5,1]=>22
[6,4,2,5,1,3]=>18
[6,4,2,5,3,1]=>20
[6,4,3,1,2,5]=>16
[6,4,3,1,5,2]=>19
[6,4,3,2,1,5]=>17
[6,4,3,2,5,1]=>21
[6,4,3,5,1,2]=>17
[6,4,3,5,2,1]=>18
[6,4,5,1,2,3]=>15
[6,4,5,1,3,2]=>16
[6,4,5,2,1,3]=>16
[6,4,5,2,3,1]=>18
[6,4,5,3,1,2]=>16
[6,4,5,3,2,1]=>17
[6,5,1,2,3,4]=>13
[6,5,1,2,4,3]=>14
[6,5,1,3,2,4]=>14
[6,5,1,3,4,2]=>16
[6,5,1,4,2,3]=>14
[6,5,1,4,3,2]=>15
[6,5,2,1,3,4]=>14
[6,5,2,1,4,3]=>15
[6,5,2,3,1,4]=>16
[6,5,2,3,4,1]=>19
[6,5,2,4,1,3]=>16
[6,5,2,4,3,1]=>18
[6,5,3,1,2,4]=>14
[6,5,3,1,4,2]=>16
[6,5,3,2,1,4]=>15
[6,5,3,2,4,1]=>18
[6,5,3,4,1,2]=>15
[6,5,3,4,2,1]=>16
[6,5,4,1,2,3]=>14
[6,5,4,1,3,2]=>15
[6,5,4,2,1,3]=>15
[6,5,4,2,3,1]=>17
[6,5,4,3,1,2]=>15
[6,5,4,3,2,1]=>16
[1,2,3,4,5,6,7]=>0
[1,2,3,4,5,7,6]=>1
[1,2,3,4,6,5,7]=>1
[1,2,3,4,6,7,5]=>3
[1,2,3,4,7,5,6]=>3
[1,2,3,4,7,6,5]=>4
[1,2,3,5,4,6,7]=>1
[1,2,3,5,4,7,6]=>2
[1,2,3,5,6,4,7]=>3
[1,2,3,5,6,7,4]=>6
[1,2,3,5,7,4,6]=>5
[1,2,3,5,7,6,4]=>7
[1,2,3,6,4,5,7]=>3
[1,2,3,6,4,7,5]=>5
[1,2,3,6,5,4,7]=>4
[1,2,3,6,5,7,4]=>7
[1,2,3,6,7,4,5]=>6
[1,2,3,6,7,5,4]=>7
[1,2,3,7,4,5,6]=>6
[1,2,3,7,4,6,5]=>7
[1,2,3,7,5,4,6]=>7
[1,2,3,7,5,6,4]=>9
[1,2,3,7,6,4,5]=>7
[1,2,3,7,6,5,4]=>8
[1,2,4,3,5,6,7]=>1
[1,2,4,3,5,7,6]=>2
[1,2,4,3,6,5,7]=>2
[1,2,4,3,6,7,5]=>4
[1,2,4,3,7,5,6]=>4
[1,2,4,3,7,6,5]=>5
[1,2,4,5,3,6,7]=>3
[1,2,4,5,3,7,6]=>4
[1,2,4,5,6,3,7]=>6
[1,2,4,5,6,7,3]=>10
[1,2,4,5,7,3,6]=>8
[1,2,4,5,7,6,3]=>11
[1,2,4,6,3,5,7]=>5
[1,2,4,6,3,7,5]=>7
[1,2,4,6,5,3,7]=>7
[1,2,4,6,5,7,3]=>11
[1,2,4,6,7,3,5]=>9
[1,2,4,6,7,5,3]=>11
[1,2,4,7,3,5,6]=>8
[1,2,4,7,3,6,5]=>9
[1,2,4,7,5,3,6]=>10
[1,2,4,7,5,6,3]=>13
[1,2,4,7,6,3,5]=>10
[1,2,4,7,6,5,3]=>12
[1,2,5,3,4,6,7]=>3
[1,2,5,3,4,7,6]=>4
[1,2,5,3,6,4,7]=>5
[1,2,5,3,6,7,4]=>8
[1,2,5,3,7,4,6]=>7
[1,2,5,3,7,6,4]=>9
[1,2,5,4,3,6,7]=>4
[1,2,5,4,3,7,6]=>5
[1,2,5,4,6,3,7]=>7
[1,2,5,4,6,7,3]=>11
[1,2,5,4,7,3,6]=>9
[1,2,5,4,7,6,3]=>12
[1,2,5,6,3,4,7]=>6
[1,2,5,6,3,7,4]=>9
[1,2,5,6,4,3,7]=>7
[1,2,5,6,4,7,3]=>11
[1,2,5,6,7,3,4]=>9
[1,2,5,6,7,4,3]=>10
[1,2,5,7,3,4,6]=>9
[1,2,5,7,3,6,4]=>11
[1,2,5,7,4,3,6]=>10
[1,2,5,7,4,6,3]=>13
[1,2,5,7,6,3,4]=>10
[1,2,5,7,6,4,3]=>11
[1,2,6,3,4,5,7]=>6
[1,2,6,3,4,7,5]=>8
[1,2,6,3,5,4,7]=>7
[1,2,6,3,5,7,4]=>10
[1,2,6,3,7,4,5]=>9
[1,2,6,3,7,5,4]=>10
[1,2,6,4,3,5,7]=>7
[1,2,6,4,3,7,5]=>9
[1,2,6,4,5,3,7]=>9
[1,2,6,4,5,7,3]=>13
[1,2,6,4,7,3,5]=>11
[1,2,6,4,7,5,3]=>13
[1,2,6,5,3,4,7]=>7
[1,2,6,5,3,7,4]=>10
[1,2,6,5,4,3,7]=>8
[1,2,6,5,4,7,3]=>12
[1,2,6,5,7,3,4]=>10
[1,2,6,5,7,4,3]=>11
[1,2,6,7,3,4,5]=>9
[1,2,6,7,3,5,4]=>10
[1,2,6,7,4,3,5]=>10
[1,2,6,7,4,5,3]=>12
[1,2,6,7,5,3,4]=>10
[1,2,6,7,5,4,3]=>11
[1,2,7,3,4,5,6]=>10
[1,2,7,3,4,6,5]=>11
[1,2,7,3,5,4,6]=>11
[1,2,7,3,5,6,4]=>13
[1,2,7,3,6,4,5]=>11
[1,2,7,3,6,5,4]=>12
[1,2,7,4,3,5,6]=>11
[1,2,7,4,3,6,5]=>12
[1,2,7,4,5,3,6]=>13
[1,2,7,4,5,6,3]=>16
[1,2,7,4,6,3,5]=>13
[1,2,7,4,6,5,3]=>15
[1,2,7,5,3,4,6]=>11
[1,2,7,5,3,6,4]=>13
[1,2,7,5,4,3,6]=>12
[1,2,7,5,4,6,3]=>15
[1,2,7,5,6,3,4]=>12
[1,2,7,5,6,4,3]=>13
[1,2,7,6,3,4,5]=>10
[1,2,7,6,3,5,4]=>11
[1,2,7,6,4,3,5]=>11
[1,2,7,6,4,5,3]=>13
[1,2,7,6,5,3,4]=>11
[1,2,7,6,5,4,3]=>12
[1,3,2,4,5,6,7]=>1
[1,3,2,4,5,7,6]=>2
[1,3,2,4,6,5,7]=>2
[1,3,2,4,6,7,5]=>4
[1,3,2,4,7,5,6]=>4
[1,3,2,4,7,6,5]=>5
[1,3,2,5,4,6,7]=>2
[1,3,2,5,4,7,6]=>3
[1,3,2,5,6,4,7]=>4
[1,3,2,5,6,7,4]=>7
[1,3,2,5,7,4,6]=>6
[1,3,2,5,7,6,4]=>8
[1,3,2,6,4,5,7]=>4
[1,3,2,6,4,7,5]=>6
[1,3,2,6,5,4,7]=>5
[1,3,2,6,5,7,4]=>8
[1,3,2,6,7,4,5]=>7
[1,3,2,6,7,5,4]=>8
[1,3,2,7,4,5,6]=>7
[1,3,2,7,4,6,5]=>8
[1,3,2,7,5,4,6]=>8
[1,3,2,7,5,6,4]=>10
[1,3,2,7,6,4,5]=>8
[1,3,2,7,6,5,4]=>9
[1,3,4,2,5,6,7]=>3
[1,3,4,2,5,7,6]=>4
[1,3,4,2,6,5,7]=>4
[1,3,4,2,6,7,5]=>6
[1,3,4,2,7,5,6]=>6
[1,3,4,2,7,6,5]=>7
[1,3,4,5,2,6,7]=>6
[1,3,4,5,2,7,6]=>7
[1,3,4,5,6,2,7]=>10
[1,3,4,5,6,7,2]=>15
[1,3,4,5,7,2,6]=>12
[1,3,4,5,7,6,2]=>16
[1,3,4,6,2,5,7]=>8
[1,3,4,6,2,7,5]=>10
[1,3,4,6,5,2,7]=>11
[1,3,4,6,5,7,2]=>16
[1,3,4,6,7,2,5]=>13
[1,3,4,6,7,5,2]=>16
[1,3,4,7,2,5,6]=>11
[1,3,4,7,2,6,5]=>12
[1,3,4,7,5,2,6]=>14
[1,3,4,7,5,6,2]=>18
[1,3,4,7,6,2,5]=>14
[1,3,4,7,6,5,2]=>17
[1,3,5,2,4,6,7]=>5
[1,3,5,2,4,7,6]=>6
[1,3,5,2,6,4,7]=>7
[1,3,5,2,6,7,4]=>10
[1,3,5,2,7,4,6]=>9
[1,3,5,2,7,6,4]=>11
[1,3,5,4,2,6,7]=>7
[1,3,5,4,2,7,6]=>8
[1,3,5,4,6,2,7]=>11
[1,3,5,4,6,7,2]=>16
[1,3,5,4,7,2,6]=>13
[1,3,5,4,7,6,2]=>17
[1,3,5,6,2,4,7]=>9
[1,3,5,6,2,7,4]=>12
[1,3,5,6,4,2,7]=>11
[1,3,5,6,4,7,2]=>16
[1,3,5,6,7,2,4]=>13
[1,3,5,6,7,4,2]=>15
[1,3,5,7,2,4,6]=>12
[1,3,5,7,2,6,4]=>14
[1,3,5,7,4,2,6]=>14
[1,3,5,7,4,6,2]=>18
[1,3,5,7,6,2,4]=>14
[1,3,5,7,6,4,2]=>16
[1,3,6,2,4,5,7]=>8
[1,3,6,2,4,7,5]=>10
[1,3,6,2,5,4,7]=>9
[1,3,6,2,5,7,4]=>12
[1,3,6,2,7,4,5]=>11
[1,3,6,2,7,5,4]=>12
[1,3,6,4,2,5,7]=>10
[1,3,6,4,2,7,5]=>12
[1,3,6,4,5,2,7]=>13
[1,3,6,4,5,7,2]=>18
[1,3,6,4,7,2,5]=>15
[1,3,6,4,7,5,2]=>18
[1,3,6,5,2,4,7]=>10
[1,3,6,5,2,7,4]=>13
[1,3,6,5,4,2,7]=>12
[1,3,6,5,4,7,2]=>17
[1,3,6,5,7,2,4]=>14
[1,3,6,5,7,4,2]=>16
[1,3,6,7,2,4,5]=>12
[1,3,6,7,2,5,4]=>13
[1,3,6,7,4,2,5]=>14
[1,3,6,7,4,5,2]=>17
[1,3,6,7,5,2,4]=>14
[1,3,6,7,5,4,2]=>16
[1,3,7,2,4,5,6]=>12
[1,3,7,2,4,6,5]=>13
[1,3,7,2,5,4,6]=>13
[1,3,7,2,5,6,4]=>15
[1,3,7,2,6,4,5]=>13
[1,3,7,2,6,5,4]=>14
[1,3,7,4,2,5,6]=>14
[1,3,7,4,2,6,5]=>15
[1,3,7,4,5,2,6]=>17
[1,3,7,4,5,6,2]=>21
[1,3,7,4,6,2,5]=>17
[1,3,7,4,6,5,2]=>20
[1,3,7,5,2,4,6]=>14
[1,3,7,5,2,6,4]=>16
[1,3,7,5,4,2,6]=>16
[1,3,7,5,4,6,2]=>20
[1,3,7,5,6,2,4]=>16
[1,3,7,5,6,4,2]=>18
[1,3,7,6,2,4,5]=>13
[1,3,7,6,2,5,4]=>14
[1,3,7,6,4,2,5]=>15
[1,3,7,6,4,5,2]=>18
[1,3,7,6,5,2,4]=>15
[1,3,7,6,5,4,2]=>17
[1,4,2,3,5,6,7]=>3
[1,4,2,3,5,7,6]=>4
[1,4,2,3,6,5,7]=>4
[1,4,2,3,6,7,5]=>6
[1,4,2,3,7,5,6]=>6
[1,4,2,3,7,6,5]=>7
[1,4,2,5,3,6,7]=>5
[1,4,2,5,3,7,6]=>6
[1,4,2,5,6,3,7]=>8
[1,4,2,5,6,7,3]=>12
[1,4,2,5,7,3,6]=>10
[1,4,2,5,7,6,3]=>13
[1,4,2,6,3,5,7]=>7
[1,4,2,6,3,7,5]=>9
[1,4,2,6,5,3,7]=>9
[1,4,2,6,5,7,3]=>13
[1,4,2,6,7,3,5]=>11
[1,4,2,6,7,5,3]=>13
[1,4,2,7,3,5,6]=>10
[1,4,2,7,3,6,5]=>11
[1,4,2,7,5,3,6]=>12
[1,4,2,7,5,6,3]=>15
[1,4,2,7,6,3,5]=>12
[1,4,2,7,6,5,3]=>14
[1,4,3,2,5,6,7]=>4
[1,4,3,2,5,7,6]=>5
[1,4,3,2,6,5,7]=>5
[1,4,3,2,6,7,5]=>7
[1,4,3,2,7,5,6]=>7
[1,4,3,2,7,6,5]=>8
[1,4,3,5,2,6,7]=>7
[1,4,3,5,2,7,6]=>8
[1,4,3,5,6,2,7]=>11
[1,4,3,5,6,7,2]=>16
[1,4,3,5,7,2,6]=>13
[1,4,3,5,7,6,2]=>17
[1,4,3,6,2,5,7]=>9
[1,4,3,6,2,7,5]=>11
[1,4,3,6,5,2,7]=>12
[1,4,3,6,5,7,2]=>17
[1,4,3,6,7,2,5]=>14
[1,4,3,6,7,5,2]=>17
[1,4,3,7,2,5,6]=>12
[1,4,3,7,2,6,5]=>13
[1,4,3,7,5,2,6]=>15
[1,4,3,7,5,6,2]=>19
[1,4,3,7,6,2,5]=>15
[1,4,3,7,6,5,2]=>18
[1,4,5,2,3,6,7]=>6
[1,4,5,2,3,7,6]=>7
[1,4,5,2,6,3,7]=>9
[1,4,5,2,6,7,3]=>13
[1,4,5,2,7,3,6]=>11
[1,4,5,2,7,6,3]=>14
[1,4,5,3,2,6,7]=>7
[1,4,5,3,2,7,6]=>8
[1,4,5,3,6,2,7]=>11
[1,4,5,3,6,7,2]=>16
[1,4,5,3,7,2,6]=>13
[1,4,5,3,7,6,2]=>17
[1,4,5,6,2,3,7]=>9
[1,4,5,6,2,7,3]=>13
[1,4,5,6,3,2,7]=>10
[1,4,5,6,3,7,2]=>15
[1,4,5,6,7,2,3]=>12
[1,4,5,6,7,3,2]=>13
[1,4,5,7,2,3,6]=>12
[1,4,5,7,2,6,3]=>15
[1,4,5,7,3,2,6]=>13
[1,4,5,7,3,6,2]=>17
[1,4,5,7,6,2,3]=>13
[1,4,5,7,6,3,2]=>14
[1,4,6,2,3,5,7]=>9
[1,4,6,2,3,7,5]=>11
[1,4,6,2,5,3,7]=>11
[1,4,6,2,5,7,3]=>15
[1,4,6,2,7,3,5]=>13
[1,4,6,2,7,5,3]=>15
[1,4,6,3,2,5,7]=>10
[1,4,6,3,2,7,5]=>12
[1,4,6,3,5,2,7]=>13
[1,4,6,3,5,7,2]=>18
[1,4,6,3,7,2,5]=>15
[1,4,6,3,7,5,2]=>18
[1,4,6,5,2,3,7]=>10
[1,4,6,5,2,7,3]=>14
[1,4,6,5,3,2,7]=>11
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Description
Number of minimal entries in the Bruhat order matrix of a permutation.
Associate to a permutation $\sigma$ of length $n$ the $n \times n$ matrix with entries
$$r_{ij}(\sigma) = \left| \big\{ u \in \{1,\dots,i\} \mid \sigma(u) \leq j \big\}\right|.$$
For the identity permutation, one has $r_{ij} = \min\{i,j\}$, and $\sigma \leq \tau$ in the (strong) Bruhat order if and only if $r_{ij}(\tau) \leq r_{ij}(\sigma)$ for all $i,j$.
This statistic records the number of indices $i,j$ with $r_{ij} = \min\{i,j\}$.
Associate to a permutation $\sigma$ of length $n$ the $n \times n$ matrix with entries
$$r_{ij}(\sigma) = \left| \big\{ u \in \{1,\dots,i\} \mid \sigma(u) \leq j \big\}\right|.$$
For the identity permutation, one has $r_{ij} = \min\{i,j\}$, and $\sigma \leq \tau$ in the (strong) Bruhat order if and only if $r_{ij}(\tau) \leq r_{ij}(\sigma)$ for all $i,j$.
This statistic records the number of indices $i,j$ with $r_{ij} = \min\{i,j\}$.
Code
def B(sigma):
n = len(sigma)
return Matrix([[sum(1 for u in [1 .. i] if sigma(u) <= j) for i in [1 .. n]] for j in [1 .. n]])
def statistic(sigma):
n = len(sigma)
M = B(sigma)
return sum(1 for i in range(n) for j in range(n) if M[i,j] == min(i,j))
Created
Jun 06, 2019 at 13:50 by Christian Stump
Updated
Jun 06, 2019 at 13:50 by Christian Stump
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