Identifier
- St001464: Permutations ⟶ ℤ
Values
=>
[1]=>1
[1,2]=>1
[2,1]=>2
[1,2,3]=>1
[1,3,2]=>2
[2,1,3]=>2
[2,3,1]=>3
[3,1,2]=>3
[3,2,1]=>2
[1,2,3,4]=>1
[1,2,4,3]=>2
[1,3,2,4]=>2
[1,3,4,2]=>3
[1,4,2,3]=>3
[1,4,3,2]=>2
[2,1,3,4]=>2
[2,1,4,3]=>4
[2,3,1,4]=>3
[2,3,4,1]=>4
[2,4,1,3]=>5
[2,4,3,1]=>3
[3,1,2,4]=>3
[3,1,4,2]=>5
[3,2,1,4]=>2
[3,2,4,1]=>3
[3,4,1,2]=>6
[3,4,2,1]=>5
[4,1,2,3]=>4
[4,1,3,2]=>3
[4,2,1,3]=>3
[4,2,3,1]=>2
[4,3,1,2]=>5
[4,3,2,1]=>4
[1,2,3,4,5]=>1
[1,2,3,5,4]=>2
[1,2,4,3,5]=>2
[1,2,4,5,3]=>3
[1,2,5,3,4]=>3
[1,2,5,4,3]=>2
[1,3,2,4,5]=>2
[1,3,2,5,4]=>4
[1,3,4,2,5]=>3
[1,3,4,5,2]=>4
[1,3,5,2,4]=>5
[1,3,5,4,2]=>3
[1,4,2,3,5]=>3
[1,4,2,5,3]=>5
[1,4,3,2,5]=>2
[1,4,3,5,2]=>3
[1,4,5,2,3]=>6
[1,4,5,3,2]=>5
[1,5,2,3,4]=>4
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>2
[1,5,4,2,3]=>5
[1,5,4,3,2]=>4
[2,1,3,4,5]=>2
[2,1,3,5,4]=>4
[2,1,4,3,5]=>4
[2,1,4,5,3]=>6
[2,1,5,3,4]=>6
[2,1,5,4,3]=>4
[2,3,1,4,5]=>3
[2,3,1,5,4]=>6
[2,3,4,1,5]=>4
[2,3,4,5,1]=>5
[2,3,5,1,4]=>7
[2,3,5,4,1]=>4
[2,4,1,3,5]=>5
[2,4,1,5,3]=>8
[2,4,3,1,5]=>3
[2,4,3,5,1]=>4
[2,4,5,1,3]=>9
[2,4,5,3,1]=>7
[2,5,1,3,4]=>7
[2,5,1,4,3]=>5
[2,5,3,1,4]=>5
[2,5,3,4,1]=>3
[2,5,4,1,3]=>8
[2,5,4,3,1]=>6
[3,1,2,4,5]=>3
[3,1,2,5,4]=>6
[3,1,4,2,5]=>5
[3,1,4,5,2]=>7
[3,1,5,2,4]=>8
[3,1,5,4,2]=>5
[3,2,1,4,5]=>2
[3,2,1,5,4]=>4
[3,2,4,1,5]=>3
[3,2,4,5,1]=>4
[3,2,5,1,4]=>5
[3,2,5,4,1]=>3
[3,4,1,2,5]=>6
[3,4,1,5,2]=>9
[3,4,2,1,5]=>5
[3,4,2,5,1]=>7
[3,4,5,1,2]=>10
[3,4,5,2,1]=>9
[3,5,1,2,4]=>9
[3,5,1,4,2]=>6
[3,5,2,1,4]=>8
[3,5,2,4,1]=>5
[3,5,4,1,2]=>9
[3,5,4,2,1]=>8
[4,1,2,3,5]=>4
[4,1,2,5,3]=>7
[4,1,3,2,5]=>3
[4,1,3,5,2]=>5
[4,1,5,2,3]=>9
[4,1,5,3,2]=>8
[4,2,1,3,5]=>3
[4,2,1,5,3]=>5
[4,2,3,1,5]=>2
[4,2,3,5,1]=>3
[4,2,5,1,3]=>6
[4,2,5,3,1]=>5
[4,3,1,2,5]=>5
[4,3,1,5,2]=>8
[4,3,2,1,5]=>4
[4,3,2,5,1]=>6
[4,3,5,1,2]=>9
[4,3,5,2,1]=>8
[4,5,1,2,3]=>10
[4,5,1,3,2]=>9
[4,5,2,1,3]=>9
[4,5,2,3,1]=>7
[4,5,3,1,2]=>6
[4,5,3,2,1]=>5
[5,1,2,3,4]=>5
[5,1,2,4,3]=>4
[5,1,3,2,4]=>4
[5,1,3,4,2]=>3
[5,1,4,2,3]=>7
[5,1,4,3,2]=>6
[5,2,1,3,4]=>4
[5,2,1,4,3]=>3
[5,2,3,1,4]=>3
[5,2,3,4,1]=>2
[5,2,4,1,3]=>5
[5,2,4,3,1]=>4
[5,3,1,2,4]=>7
[5,3,1,4,2]=>5
[5,3,2,1,4]=>6
[5,3,2,4,1]=>4
[5,3,4,1,2]=>7
[5,3,4,2,1]=>6
[5,4,1,2,3]=>9
[5,4,1,3,2]=>8
[5,4,2,1,3]=>8
[5,4,2,3,1]=>6
[5,4,3,1,2]=>5
[5,4,3,2,1]=>4
[1,2,3,4,5,6]=>1
[1,2,3,4,6,5]=>2
[1,2,3,5,4,6]=>2
[1,2,3,5,6,4]=>3
[1,2,3,6,4,5]=>3
[1,2,3,6,5,4]=>2
[1,2,4,3,5,6]=>2
[1,2,4,3,6,5]=>4
[1,2,4,5,3,6]=>3
[1,2,4,5,6,3]=>4
[1,2,4,6,3,5]=>5
[1,2,4,6,5,3]=>3
[1,2,5,3,4,6]=>3
[1,2,5,3,6,4]=>5
[1,2,5,4,3,6]=>2
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>6
[1,2,5,6,4,3]=>5
[1,2,6,3,4,5]=>4
[1,2,6,3,5,4]=>3
[1,2,6,4,3,5]=>3
[1,2,6,4,5,3]=>2
[1,2,6,5,3,4]=>5
[1,2,6,5,4,3]=>4
[1,3,2,4,5,6]=>2
[1,3,2,4,6,5]=>4
[1,3,2,5,4,6]=>4
[1,3,2,5,6,4]=>6
[1,3,2,6,4,5]=>6
[1,3,2,6,5,4]=>4
[1,3,4,2,5,6]=>3
[1,3,4,2,6,5]=>6
[1,3,4,5,2,6]=>4
[1,3,4,5,6,2]=>5
[1,3,4,6,2,5]=>7
[1,3,4,6,5,2]=>4
[1,3,5,2,4,6]=>5
[1,3,5,2,6,4]=>8
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>4
[1,3,5,6,2,4]=>9
[1,3,5,6,4,2]=>7
[1,3,6,2,4,5]=>7
[1,3,6,2,5,4]=>5
[1,3,6,4,2,5]=>5
[1,3,6,4,5,2]=>3
[1,3,6,5,2,4]=>8
[1,3,6,5,4,2]=>6
[1,4,2,3,5,6]=>3
[1,4,2,3,6,5]=>6
[1,4,2,5,3,6]=>5
[1,4,2,5,6,3]=>7
[1,4,2,6,3,5]=>8
[1,4,2,6,5,3]=>5
[1,4,3,2,5,6]=>2
[1,4,3,2,6,5]=>4
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>4
[1,4,3,6,2,5]=>5
[1,4,3,6,5,2]=>3
[1,4,5,2,3,6]=>6
[1,4,5,2,6,3]=>9
[1,4,5,3,2,6]=>5
[1,4,5,3,6,2]=>7
[1,4,5,6,2,3]=>10
[1,4,5,6,3,2]=>9
[1,4,6,2,3,5]=>9
[1,4,6,2,5,3]=>6
[1,4,6,3,2,5]=>8
[1,4,6,3,5,2]=>5
[1,4,6,5,2,3]=>9
[1,4,6,5,3,2]=>8
[1,5,2,3,4,6]=>4
[1,5,2,3,6,4]=>7
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>5
[1,5,2,6,3,4]=>9
[1,5,2,6,4,3]=>8
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>5
[1,5,3,4,2,6]=>2
[1,5,3,4,6,2]=>3
[1,5,3,6,2,4]=>6
[1,5,3,6,4,2]=>5
[1,5,4,2,3,6]=>5
[1,5,4,2,6,3]=>8
[1,5,4,3,2,6]=>4
[1,5,4,3,6,2]=>6
[1,5,4,6,2,3]=>9
[1,5,4,6,3,2]=>8
[1,5,6,2,3,4]=>10
[1,5,6,2,4,3]=>9
[1,5,6,3,2,4]=>9
[1,5,6,3,4,2]=>7
[1,5,6,4,2,3]=>6
[1,5,6,4,3,2]=>5
[1,6,2,3,4,5]=>5
[1,6,2,3,5,4]=>4
[1,6,2,4,3,5]=>4
[1,6,2,4,5,3]=>3
[1,6,2,5,3,4]=>7
[1,6,2,5,4,3]=>6
[1,6,3,2,4,5]=>4
[1,6,3,2,5,4]=>3
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>2
[1,6,3,5,2,4]=>5
[1,6,3,5,4,2]=>4
[1,6,4,2,3,5]=>7
[1,6,4,2,5,3]=>5
[1,6,4,3,2,5]=>6
[1,6,4,3,5,2]=>4
[1,6,4,5,2,3]=>7
[1,6,4,5,3,2]=>6
[1,6,5,2,3,4]=>9
[1,6,5,2,4,3]=>8
[1,6,5,3,2,4]=>8
[1,6,5,3,4,2]=>6
[1,6,5,4,2,3]=>5
[1,6,5,4,3,2]=>4
[2,1,3,4,5,6]=>2
[2,1,3,4,6,5]=>4
[2,1,3,5,4,6]=>4
[2,1,3,5,6,4]=>6
[2,1,3,6,4,5]=>6
[2,1,3,6,5,4]=>4
[2,1,4,3,5,6]=>4
[2,1,4,3,6,5]=>8
[2,1,4,5,3,6]=>6
[2,1,4,5,6,3]=>8
[2,1,4,6,3,5]=>10
[2,1,4,6,5,3]=>6
[2,1,5,3,4,6]=>6
[2,1,5,3,6,4]=>10
[2,1,5,4,3,6]=>4
[2,1,5,4,6,3]=>6
[2,1,5,6,3,4]=>12
[2,1,5,6,4,3]=>10
[2,1,6,3,4,5]=>8
[2,1,6,3,5,4]=>6
[2,1,6,4,3,5]=>6
[2,1,6,4,5,3]=>4
[2,1,6,5,3,4]=>10
[2,1,6,5,4,3]=>8
[2,3,1,4,5,6]=>3
[2,3,1,4,6,5]=>6
[2,3,1,5,4,6]=>6
[2,3,1,5,6,4]=>9
[2,3,1,6,4,5]=>9
[2,3,1,6,5,4]=>6
[2,3,4,1,5,6]=>4
[2,3,4,1,6,5]=>8
[2,3,4,5,1,6]=>5
[2,3,4,5,6,1]=>6
[2,3,4,6,1,5]=>9
[2,3,4,6,5,1]=>5
[2,3,5,1,4,6]=>7
[2,3,5,1,6,4]=>11
[2,3,5,4,1,6]=>4
[2,3,5,4,6,1]=>5
[2,3,5,6,1,4]=>12
[2,3,5,6,4,1]=>9
[2,3,6,1,4,5]=>10
[2,3,6,1,5,4]=>7
[2,3,6,4,1,5]=>7
[2,3,6,4,5,1]=>4
[2,3,6,5,1,4]=>11
[2,3,6,5,4,1]=>8
[2,4,1,3,5,6]=>5
[2,4,1,3,6,5]=>10
[2,4,1,5,3,6]=>8
[2,4,1,5,6,3]=>11
[2,4,1,6,3,5]=>13
[2,4,1,6,5,3]=>8
[2,4,3,1,5,6]=>3
[2,4,3,1,6,5]=>6
[2,4,3,5,1,6]=>4
[2,4,3,5,6,1]=>5
[2,4,3,6,1,5]=>7
[2,4,3,6,5,1]=>4
[2,4,5,1,3,6]=>9
[2,4,5,1,6,3]=>13
[2,4,5,3,1,6]=>7
[2,4,5,3,6,1]=>9
[2,4,5,6,1,3]=>14
[2,4,5,6,3,1]=>12
[2,4,6,1,3,5]=>14
[2,4,6,1,5,3]=>9
[2,4,6,3,1,5]=>12
[2,4,6,3,5,1]=>7
[2,4,6,5,1,3]=>13
[2,4,6,5,3,1]=>11
[2,5,1,3,4,6]=>7
[2,5,1,3,6,4]=>12
[2,5,1,4,3,6]=>5
[2,5,1,4,6,3]=>8
[2,5,1,6,3,4]=>15
[2,5,1,6,4,3]=>13
[2,5,3,1,4,6]=>5
[2,5,3,1,6,4]=>8
[2,5,3,4,1,6]=>3
[2,5,3,4,6,1]=>4
[2,5,3,6,1,4]=>9
[2,5,3,6,4,1]=>7
[2,5,4,1,3,6]=>8
[2,5,4,1,6,3]=>12
[2,5,4,3,1,6]=>6
[2,5,4,3,6,1]=>8
[2,5,4,6,1,3]=>13
[2,5,4,6,3,1]=>11
[2,5,6,1,3,4]=>16
[2,5,6,1,4,3]=>14
[2,5,6,3,1,4]=>14
[2,5,6,3,4,1]=>10
[2,5,6,4,1,3]=>9
[2,5,6,4,3,1]=>7
[2,6,1,3,4,5]=>9
[2,6,1,3,5,4]=>7
[2,6,1,4,3,5]=>7
[2,6,1,4,5,3]=>5
[2,6,1,5,3,4]=>12
[2,6,1,5,4,3]=>10
[2,6,3,1,4,5]=>7
[2,6,3,1,5,4]=>5
[2,6,3,4,1,5]=>5
[2,6,3,4,5,1]=>3
[2,6,3,5,1,4]=>8
[2,6,3,5,4,1]=>6
[2,6,4,1,3,5]=>12
[2,6,4,1,5,3]=>8
[2,6,4,3,1,5]=>10
[2,6,4,3,5,1]=>6
[2,6,4,5,1,3]=>11
[2,6,4,5,3,1]=>9
[2,6,5,1,3,4]=>15
[2,6,5,1,4,3]=>13
[2,6,5,3,1,4]=>13
[2,6,5,3,4,1]=>9
[2,6,5,4,1,3]=>8
[2,6,5,4,3,1]=>6
[3,1,2,4,5,6]=>3
[3,1,2,4,6,5]=>6
[3,1,2,5,4,6]=>6
[3,1,2,5,6,4]=>9
[3,1,2,6,4,5]=>9
[3,1,2,6,5,4]=>6
[3,1,4,2,5,6]=>5
[3,1,4,2,6,5]=>10
[3,1,4,5,2,6]=>7
[3,1,4,5,6,2]=>9
[3,1,4,6,2,5]=>12
[3,1,4,6,5,2]=>7
[3,1,5,2,4,6]=>8
[3,1,5,2,6,4]=>13
[3,1,5,4,2,6]=>5
[3,1,5,4,6,2]=>7
[3,1,5,6,2,4]=>15
[3,1,5,6,4,2]=>12
[3,1,6,2,4,5]=>11
[3,1,6,2,5,4]=>8
[3,1,6,4,2,5]=>8
[3,1,6,4,5,2]=>5
[3,1,6,5,2,4]=>13
[3,1,6,5,4,2]=>10
[3,2,1,4,5,6]=>2
[3,2,1,4,6,5]=>4
[3,2,1,5,4,6]=>4
[3,2,1,5,6,4]=>6
[3,2,1,6,4,5]=>6
[3,2,1,6,5,4]=>4
[3,2,4,1,5,6]=>3
[3,2,4,1,6,5]=>6
[3,2,4,5,1,6]=>4
[3,2,4,5,6,1]=>5
[3,2,4,6,1,5]=>7
[3,2,4,6,5,1]=>4
[3,2,5,1,4,6]=>5
[3,2,5,1,6,4]=>8
[3,2,5,4,1,6]=>3
[3,2,5,4,6,1]=>4
[3,2,5,6,1,4]=>9
[3,2,5,6,4,1]=>7
[3,2,6,1,4,5]=>7
[3,2,6,1,5,4]=>5
[3,2,6,4,1,5]=>5
[3,2,6,4,5,1]=>3
[3,2,6,5,1,4]=>8
[3,2,6,5,4,1]=>6
[3,4,1,2,5,6]=>6
[3,4,1,2,6,5]=>12
[3,4,1,5,2,6]=>9
[3,4,1,5,6,2]=>12
[3,4,1,6,2,5]=>15
[3,4,1,6,5,2]=>9
[3,4,2,1,5,6]=>5
[3,4,2,1,6,5]=>10
[3,4,2,5,1,6]=>7
[3,4,2,5,6,1]=>9
[3,4,2,6,1,5]=>12
[3,4,2,6,5,1]=>7
[3,4,5,1,2,6]=>10
[3,4,5,1,6,2]=>14
[3,4,5,2,1,6]=>9
[3,4,5,2,6,1]=>12
[3,4,5,6,1,2]=>15
[3,4,5,6,2,1]=>14
[3,4,6,1,2,5]=>16
[3,4,6,1,5,2]=>10
[3,4,6,2,1,5]=>15
[3,4,6,2,5,1]=>9
[3,4,6,5,1,2]=>14
[3,4,6,5,2,1]=>13
[3,5,1,2,4,6]=>9
[3,5,1,2,6,4]=>15
[3,5,1,4,2,6]=>6
[3,5,1,4,6,2]=>9
[3,5,1,6,2,4]=>18
[3,5,1,6,4,2]=>15
[3,5,2,1,4,6]=>8
[3,5,2,1,6,4]=>13
[3,5,2,4,1,6]=>5
[3,5,2,4,6,1]=>7
[3,5,2,6,1,4]=>15
[3,5,2,6,4,1]=>12
[3,5,4,1,2,6]=>9
[3,5,4,1,6,2]=>13
[3,5,4,2,1,6]=>8
[3,5,4,2,6,1]=>11
[3,5,4,6,1,2]=>14
[3,5,4,6,2,1]=>13
[3,5,6,1,2,4]=>19
[3,5,6,1,4,2]=>16
[3,5,6,2,1,4]=>18
[3,5,6,2,4,1]=>14
[3,5,6,4,1,2]=>10
[3,5,6,4,2,1]=>9
[3,6,1,2,4,5]=>12
[3,6,1,2,5,4]=>9
[3,6,1,4,2,5]=>9
[3,6,1,4,5,2]=>6
[3,6,1,5,2,4]=>15
[3,6,1,5,4,2]=>12
[3,6,2,1,4,5]=>11
[3,6,2,1,5,4]=>8
[3,6,2,4,1,5]=>8
[3,6,2,4,5,1]=>5
[3,6,2,5,1,4]=>13
[3,6,2,5,4,1]=>10
[3,6,4,1,2,5]=>14
[3,6,4,1,5,2]=>9
[3,6,4,2,1,5]=>13
[3,6,4,2,5,1]=>8
[3,6,4,5,1,2]=>12
[3,6,4,5,2,1]=>11
[3,6,5,1,2,4]=>18
[3,6,5,1,4,2]=>15
[3,6,5,2,1,4]=>17
[3,6,5,2,4,1]=>13
[3,6,5,4,1,2]=>9
[3,6,5,4,2,1]=>8
[4,1,2,3,5,6]=>4
[4,1,2,3,6,5]=>8
[4,1,2,5,3,6]=>7
[4,1,2,5,6,3]=>10
[4,1,2,6,3,5]=>11
[4,1,2,6,5,3]=>7
[4,1,3,2,5,6]=>3
[4,1,3,2,6,5]=>6
[4,1,3,5,2,6]=>5
[4,1,3,5,6,2]=>7
[4,1,3,6,2,5]=>8
[4,1,3,6,5,2]=>5
[4,1,5,2,3,6]=>9
[4,1,5,2,6,3]=>14
[4,1,5,3,2,6]=>8
[4,1,5,3,6,2]=>12
[4,1,5,6,2,3]=>16
[4,1,5,6,3,2]=>15
[4,1,6,2,3,5]=>13
[4,1,6,2,5,3]=>9
[4,1,6,3,2,5]=>12
[4,1,6,3,5,2]=>8
[4,1,6,5,2,3]=>14
[4,1,6,5,3,2]=>13
[4,2,1,3,5,6]=>3
[4,2,1,3,6,5]=>6
[4,2,1,5,3,6]=>5
[4,2,1,5,6,3]=>7
[4,2,1,6,3,5]=>8
[4,2,1,6,5,3]=>5
[4,2,3,1,5,6]=>2
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>3
[4,2,3,5,6,1]=>4
[4,2,3,6,1,5]=>5
[4,2,3,6,5,1]=>3
[4,2,5,1,3,6]=>6
[4,2,5,1,6,3]=>9
[4,2,5,3,1,6]=>5
[4,2,5,3,6,1]=>7
[4,2,5,6,1,3]=>10
[4,2,5,6,3,1]=>9
[4,2,6,1,3,5]=>9
[4,2,6,1,5,3]=>6
[4,2,6,3,1,5]=>8
[4,2,6,3,5,1]=>5
[4,2,6,5,1,3]=>9
[4,2,6,5,3,1]=>8
[4,3,1,2,5,6]=>5
[4,3,1,2,6,5]=>10
[4,3,1,5,2,6]=>8
[4,3,1,5,6,2]=>11
[4,3,1,6,2,5]=>13
[4,3,1,6,5,2]=>8
[4,3,2,1,5,6]=>4
[4,3,2,1,6,5]=>8
[4,3,2,5,1,6]=>6
[4,3,2,5,6,1]=>8
[4,3,2,6,1,5]=>10
[4,3,2,6,5,1]=>6
[4,3,5,1,2,6]=>9
[4,3,5,1,6,2]=>13
[4,3,5,2,1,6]=>8
[4,3,5,2,6,1]=>11
[4,3,5,6,1,2]=>14
[4,3,5,6,2,1]=>13
[4,3,6,1,2,5]=>14
[4,3,6,1,5,2]=>9
[4,3,6,2,1,5]=>13
[4,3,6,2,5,1]=>8
[4,3,6,5,1,2]=>13
[4,3,6,5,2,1]=>12
[4,5,1,2,3,6]=>10
[4,5,1,2,6,3]=>16
[4,5,1,3,2,6]=>9
[4,5,1,3,6,2]=>14
[4,5,1,6,2,3]=>19
[4,5,1,6,3,2]=>18
[4,5,2,1,3,6]=>9
[4,5,2,1,6,3]=>14
[4,5,2,3,1,6]=>7
[4,5,2,3,6,1]=>10
[4,5,2,6,1,3]=>16
[4,5,2,6,3,1]=>14
[4,5,3,1,2,6]=>6
[4,5,3,1,6,2]=>9
[4,5,3,2,1,6]=>5
[4,5,3,2,6,1]=>7
[4,5,3,6,1,2]=>10
[4,5,3,6,2,1]=>9
[4,5,6,1,2,3]=>20
[4,5,6,1,3,2]=>19
[4,5,6,2,1,3]=>19
[4,5,6,2,3,1]=>16
[4,5,6,3,1,2]=>16
[4,5,6,3,2,1]=>14
[4,6,1,2,3,5]=>14
[4,6,1,2,5,3]=>10
[4,6,1,3,2,5]=>13
[4,6,1,3,5,2]=>9
[4,6,1,5,2,3]=>16
[4,6,1,5,3,2]=>15
[4,6,2,1,3,5]=>13
[4,6,2,1,5,3]=>9
[4,6,2,3,1,5]=>11
[4,6,2,3,5,1]=>7
[4,6,2,5,1,3]=>14
[4,6,2,5,3,1]=>12
[4,6,3,1,2,5]=>9
[4,6,3,1,5,2]=>6
[4,6,3,2,1,5]=>8
[4,6,3,2,5,1]=>5
[4,6,3,5,1,2]=>9
[4,6,3,5,2,1]=>8
[4,6,5,1,2,3]=>19
[4,6,5,1,3,2]=>18
[4,6,5,2,1,3]=>18
[4,6,5,2,3,1]=>15
[4,6,5,3,1,2]=>15
[4,6,5,3,2,1]=>13
[5,1,2,3,4,6]=>5
[5,1,2,3,6,4]=>9
[5,1,2,4,3,6]=>4
[5,1,2,4,6,3]=>7
[5,1,2,6,3,4]=>12
[5,1,2,6,4,3]=>11
[5,1,3,2,4,6]=>4
[5,1,3,2,6,4]=>7
[5,1,3,4,2,6]=>3
[5,1,3,4,6,2]=>5
[5,1,3,6,2,4]=>9
[5,1,3,6,4,2]=>8
[5,1,4,2,3,6]=>7
[5,1,4,2,6,3]=>12
[5,1,4,3,2,6]=>6
[5,1,4,3,6,2]=>10
[5,1,4,6,2,3]=>14
[5,1,4,6,3,2]=>13
[5,1,6,2,3,4]=>14
[5,1,6,2,4,3]=>13
[5,1,6,3,2,4]=>13
[5,1,6,3,4,2]=>11
[5,1,6,4,2,3]=>9
[5,1,6,4,3,2]=>8
[5,2,1,3,4,6]=>4
[5,2,1,3,6,4]=>7
[5,2,1,4,3,6]=>3
[5,2,1,4,6,3]=>5
[5,2,1,6,3,4]=>9
[5,2,1,6,4,3]=>8
[5,2,3,1,4,6]=>3
[5,2,3,1,6,4]=>5
[5,2,3,4,1,6]=>2
[5,2,3,4,6,1]=>3
[5,2,3,6,1,4]=>6
[5,2,3,6,4,1]=>5
[5,2,4,1,3,6]=>5
[5,2,4,1,6,3]=>8
[5,2,4,3,1,6]=>4
[5,2,4,3,6,1]=>6
[5,2,4,6,1,3]=>9
[5,2,4,6,3,1]=>8
[5,2,6,1,3,4]=>10
[5,2,6,1,4,3]=>9
[5,2,6,3,1,4]=>9
[5,2,6,3,4,1]=>7
[5,2,6,4,1,3]=>6
[5,2,6,4,3,1]=>5
[5,3,1,2,4,6]=>7
[5,3,1,2,6,4]=>12
[5,3,1,4,2,6]=>5
[5,3,1,4,6,2]=>8
[5,3,1,6,2,4]=>15
[5,3,1,6,4,2]=>13
[5,3,2,1,4,6]=>6
[5,3,2,1,6,4]=>10
[5,3,2,4,1,6]=>4
[5,3,2,4,6,1]=>6
[5,3,2,6,1,4]=>12
[5,3,2,6,4,1]=>10
[5,3,4,1,2,6]=>7
[5,3,4,1,6,2]=>11
[5,3,4,2,1,6]=>6
[5,3,4,2,6,1]=>9
[5,3,4,6,1,2]=>12
[5,3,4,6,2,1]=>11
[5,3,6,1,2,4]=>16
[5,3,6,1,4,2]=>14
[5,3,6,2,1,4]=>15
[5,3,6,2,4,1]=>12
[5,3,6,4,1,2]=>9
[5,3,6,4,2,1]=>8
[5,4,1,2,3,6]=>9
[5,4,1,2,6,3]=>15
[5,4,1,3,2,6]=>8
[5,4,1,3,6,2]=>13
[5,4,1,6,2,3]=>18
[5,4,1,6,3,2]=>17
[5,4,2,1,3,6]=>8
[5,4,2,1,6,3]=>13
[5,4,2,3,1,6]=>6
[5,4,2,3,6,1]=>9
[5,4,2,6,1,3]=>15
[5,4,2,6,3,1]=>13
[5,4,3,1,2,6]=>5
[5,4,3,1,6,2]=>8
[5,4,3,2,1,6]=>4
[5,4,3,2,6,1]=>6
[5,4,3,6,1,2]=>9
[5,4,3,6,2,1]=>8
[5,4,6,1,2,3]=>19
[5,4,6,1,3,2]=>18
[5,4,6,2,1,3]=>18
[5,4,6,2,3,1]=>15
[5,4,6,3,1,2]=>15
[5,4,6,3,2,1]=>13
[5,6,1,2,3,4]=>15
[5,6,1,2,4,3]=>14
[5,6,1,3,2,4]=>14
[5,6,1,3,4,2]=>12
[5,6,1,4,2,3]=>10
[5,6,1,4,3,2]=>9
[5,6,2,1,3,4]=>14
[5,6,2,1,4,3]=>13
[5,6,2,3,1,4]=>12
[5,6,2,3,4,1]=>9
[5,6,2,4,1,3]=>9
[5,6,2,4,3,1]=>7
[5,6,3,1,2,4]=>10
[5,6,3,1,4,2]=>9
[5,6,3,2,1,4]=>9
[5,6,3,2,4,1]=>7
[5,6,3,4,1,2]=>6
[5,6,3,4,2,1]=>5
[5,6,4,1,2,3]=>16
[5,6,4,1,3,2]=>15
[5,6,4,2,1,3]=>15
[5,6,4,2,3,1]=>12
[5,6,4,3,1,2]=>12
[5,6,4,3,2,1]=>10
[6,1,2,3,4,5]=>6
[6,1,2,3,5,4]=>5
[6,1,2,4,3,5]=>5
[6,1,2,4,5,3]=>4
[6,1,2,5,3,4]=>9
[6,1,2,5,4,3]=>8
[6,1,3,2,4,5]=>5
[6,1,3,2,5,4]=>4
[6,1,3,4,2,5]=>4
[6,1,3,4,5,2]=>3
[6,1,3,5,2,4]=>7
[6,1,3,5,4,2]=>6
[6,1,4,2,3,5]=>9
[6,1,4,2,5,3]=>7
[6,1,4,3,2,5]=>8
[6,1,4,3,5,2]=>6
[6,1,4,5,2,3]=>10
[6,1,4,5,3,2]=>9
[6,1,5,2,3,4]=>12
[6,1,5,2,4,3]=>11
[6,1,5,3,2,4]=>11
[6,1,5,3,4,2]=>9
[6,1,5,4,2,3]=>7
[6,1,5,4,3,2]=>6
[6,2,1,3,4,5]=>5
[6,2,1,3,5,4]=>4
[6,2,1,4,3,5]=>4
[6,2,1,4,5,3]=>3
[6,2,1,5,3,4]=>7
[6,2,1,5,4,3]=>6
[6,2,3,1,4,5]=>4
[6,2,3,1,5,4]=>3
[6,2,3,4,1,5]=>3
[6,2,3,4,5,1]=>2
[6,2,3,5,1,4]=>5
[6,2,3,5,4,1]=>4
[6,2,4,1,3,5]=>7
[6,2,4,1,5,3]=>5
[6,2,4,3,1,5]=>6
[6,2,4,3,5,1]=>4
[6,2,4,5,1,3]=>7
[6,2,4,5,3,1]=>6
[6,2,5,1,3,4]=>9
[6,2,5,1,4,3]=>8
[6,2,5,3,1,4]=>8
[6,2,5,3,4,1]=>6
[6,2,5,4,1,3]=>5
[6,2,5,4,3,1]=>4
[6,3,1,2,4,5]=>9
[6,3,1,2,5,4]=>7
[6,3,1,4,2,5]=>7
[6,3,1,4,5,2]=>5
[6,3,1,5,2,4]=>12
[6,3,1,5,4,2]=>10
[6,3,2,1,4,5]=>8
[6,3,2,1,5,4]=>6
[6,3,2,4,1,5]=>6
[6,3,2,4,5,1]=>4
[6,3,2,5,1,4]=>10
[6,3,2,5,4,1]=>8
[6,3,4,1,2,5]=>10
[6,3,4,1,5,2]=>7
[6,3,4,2,1,5]=>9
[6,3,4,2,5,1]=>6
[6,3,4,5,1,2]=>9
[6,3,4,5,2,1]=>8
[6,3,5,1,2,4]=>14
[6,3,5,1,4,2]=>12
[6,3,5,2,1,4]=>13
[6,3,5,2,4,1]=>10
[6,3,5,4,1,2]=>7
[6,3,5,4,2,1]=>6
[6,4,1,2,3,5]=>12
[6,4,1,2,5,3]=>9
[6,4,1,3,2,5]=>11
[6,4,1,3,5,2]=>8
[6,4,1,5,2,3]=>14
[6,4,1,5,3,2]=>13
[6,4,2,1,3,5]=>11
[6,4,2,1,5,3]=>8
[6,4,2,3,1,5]=>9
[6,4,2,3,5,1]=>6
[6,4,2,5,1,3]=>12
[6,4,2,5,3,1]=>10
[6,4,3,1,2,5]=>7
[6,4,3,1,5,2]=>5
[6,4,3,2,1,5]=>6
[6,4,3,2,5,1]=>4
[6,4,3,5,1,2]=>7
[6,4,3,5,2,1]=>6
[6,4,5,1,2,3]=>16
[6,4,5,1,3,2]=>15
[6,4,5,2,1,3]=>15
[6,4,5,2,3,1]=>12
[6,4,5,3,1,2]=>12
[6,4,5,3,2,1]=>10
[6,5,1,2,3,4]=>14
[6,5,1,2,4,3]=>13
[6,5,1,3,2,4]=>13
[6,5,1,3,4,2]=>11
[6,5,1,4,2,3]=>9
[6,5,1,4,3,2]=>8
[6,5,2,1,3,4]=>13
[6,5,2,1,4,3]=>12
[6,5,2,3,1,4]=>11
[6,5,2,3,4,1]=>8
[6,5,2,4,1,3]=>8
[6,5,2,4,3,1]=>6
[6,5,3,1,2,4]=>9
[6,5,3,1,4,2]=>8
[6,5,3,2,1,4]=>8
[6,5,3,2,4,1]=>6
[6,5,3,4,1,2]=>5
[6,5,3,4,2,1]=>4
[6,5,4,1,2,3]=>14
[6,5,4,1,3,2]=>13
[6,5,4,2,1,3]=>13
[6,5,4,2,3,1]=>10
[6,5,4,3,1,2]=>10
[6,5,4,3,2,1]=>8
[1,2,3,4,5,6,7]=>1
[1,2,3,4,5,7,6]=>2
[1,2,3,4,6,5,7]=>2
[1,2,3,4,6,7,5]=>3
[1,2,3,4,7,5,6]=>3
[1,2,3,4,7,6,5]=>2
[1,2,3,5,4,6,7]=>2
[1,2,3,5,4,7,6]=>4
[1,2,3,5,6,4,7]=>3
[1,2,3,5,6,7,4]=>4
[1,2,3,5,7,4,6]=>5
[1,2,3,5,7,6,4]=>3
[1,2,3,6,4,5,7]=>3
[1,2,3,6,4,7,5]=>5
[1,2,3,6,5,4,7]=>2
[1,2,3,6,5,7,4]=>3
[1,2,3,6,7,4,5]=>6
[1,2,3,6,7,5,4]=>5
[1,2,3,7,4,5,6]=>4
[1,2,3,7,4,6,5]=>3
[1,2,3,7,5,4,6]=>3
[1,2,3,7,5,6,4]=>2
[1,2,3,7,6,4,5]=>5
[1,2,3,7,6,5,4]=>4
[1,2,4,3,5,6,7]=>2
[1,2,4,3,5,7,6]=>4
[1,2,4,3,6,5,7]=>4
[1,2,4,3,6,7,5]=>6
[1,2,4,3,7,5,6]=>6
[1,2,4,3,7,6,5]=>4
[1,2,4,5,3,6,7]=>3
[1,2,4,5,3,7,6]=>6
[1,2,4,5,6,3,7]=>4
[1,2,4,5,6,7,3]=>5
[1,2,4,5,7,3,6]=>7
[1,2,4,5,7,6,3]=>4
[1,2,4,6,3,5,7]=>5
[1,2,4,6,3,7,5]=>8
[1,2,4,6,5,3,7]=>3
[1,2,4,6,5,7,3]=>4
[1,2,4,6,7,3,5]=>9
[1,2,4,6,7,5,3]=>7
[1,2,4,7,3,5,6]=>7
[1,2,4,7,3,6,5]=>5
[1,2,4,7,5,3,6]=>5
[1,2,4,7,5,6,3]=>3
[1,2,4,7,6,3,5]=>8
[1,2,4,7,6,5,3]=>6
[1,2,5,3,4,6,7]=>3
[1,2,5,3,4,7,6]=>6
[1,2,5,3,6,4,7]=>5
[1,2,5,3,6,7,4]=>7
[1,2,5,3,7,4,6]=>8
[1,2,5,3,7,6,4]=>5
[1,2,5,4,3,6,7]=>2
[1,2,5,4,3,7,6]=>4
[1,2,5,4,6,3,7]=>3
[1,2,5,4,6,7,3]=>4
[1,2,5,4,7,3,6]=>5
[1,2,5,4,7,6,3]=>3
[1,2,5,6,3,4,7]=>6
[1,2,5,6,3,7,4]=>9
[1,2,5,6,4,3,7]=>5
[1,2,5,6,4,7,3]=>7
[1,2,5,6,7,3,4]=>10
[1,2,5,6,7,4,3]=>9
[1,2,5,7,3,4,6]=>9
[1,2,5,7,3,6,4]=>6
[1,2,5,7,4,3,6]=>8
[1,2,5,7,4,6,3]=>5
[1,2,5,7,6,3,4]=>9
[1,2,5,7,6,4,3]=>8
[1,2,6,3,4,5,7]=>4
[1,2,6,3,4,7,5]=>7
[1,2,6,3,5,4,7]=>3
[1,2,6,3,5,7,4]=>5
[1,2,6,3,7,4,5]=>9
[1,2,6,3,7,5,4]=>8
[1,2,6,4,3,5,7]=>3
[1,2,6,4,3,7,5]=>5
[1,2,6,4,5,3,7]=>2
[1,2,6,4,5,7,3]=>3
[1,2,6,4,7,3,5]=>6
[1,2,6,4,7,5,3]=>5
[1,2,6,5,3,4,7]=>5
[1,2,6,5,3,7,4]=>8
[1,2,6,5,4,3,7]=>4
[1,2,6,5,4,7,3]=>6
[1,2,6,5,7,3,4]=>9
[1,2,6,5,7,4,3]=>8
[1,2,6,7,3,4,5]=>10
[1,2,6,7,3,5,4]=>9
[1,2,6,7,4,3,5]=>9
[1,2,6,7,4,5,3]=>7
[1,2,6,7,5,3,4]=>6
[1,2,6,7,5,4,3]=>5
[1,2,7,3,4,5,6]=>5
[1,2,7,3,4,6,5]=>4
[1,2,7,3,5,4,6]=>4
[1,2,7,3,5,6,4]=>3
[1,2,7,3,6,4,5]=>7
[1,2,7,3,6,5,4]=>6
[1,2,7,4,3,5,6]=>4
[1,2,7,4,3,6,5]=>3
[1,2,7,4,5,3,6]=>3
[1,2,7,4,5,6,3]=>2
[1,2,7,4,6,3,5]=>5
[1,2,7,4,6,5,3]=>4
[1,2,7,5,3,4,6]=>7
[1,2,7,5,3,6,4]=>5
[1,2,7,5,4,3,6]=>6
[1,2,7,5,4,6,3]=>4
[1,2,7,5,6,3,4]=>7
[1,2,7,5,6,4,3]=>6
[1,2,7,6,3,4,5]=>9
[1,2,7,6,3,5,4]=>8
[1,2,7,6,4,3,5]=>8
[1,2,7,6,4,5,3]=>6
[1,2,7,6,5,3,4]=>5
[1,2,7,6,5,4,3]=>4
[1,3,2,4,5,6,7]=>2
[1,3,2,4,5,7,6]=>4
[1,3,2,4,6,5,7]=>4
[1,3,2,4,6,7,5]=>6
[1,3,2,4,7,5,6]=>6
[1,3,2,4,7,6,5]=>4
[1,3,2,5,4,6,7]=>4
[1,3,2,5,4,7,6]=>8
[1,3,2,5,6,4,7]=>6
[1,3,2,5,6,7,4]=>8
[1,3,2,5,7,4,6]=>10
[1,3,2,5,7,6,4]=>6
[1,3,2,6,4,5,7]=>6
[1,3,2,6,4,7,5]=>10
[1,3,2,6,5,4,7]=>4
[1,3,2,6,5,7,4]=>6
[1,3,2,6,7,4,5]=>12
[1,3,2,6,7,5,4]=>10
[1,3,2,7,4,5,6]=>8
[1,3,2,7,4,6,5]=>6
[1,3,2,7,5,4,6]=>6
[1,3,2,7,5,6,4]=>4
[1,3,2,7,6,4,5]=>10
[1,3,2,7,6,5,4]=>8
[1,3,4,2,5,6,7]=>3
[1,3,4,2,5,7,6]=>6
[1,3,4,2,6,5,7]=>6
[1,3,4,2,6,7,5]=>9
[1,3,4,2,7,5,6]=>9
[1,3,4,2,7,6,5]=>6
[1,3,4,5,2,6,7]=>4
[1,3,4,5,2,7,6]=>8
[1,3,4,5,6,2,7]=>5
[1,3,4,5,6,7,2]=>6
[1,3,4,5,7,2,6]=>9
[1,3,4,5,7,6,2]=>5
[1,3,4,6,2,5,7]=>7
[1,3,4,6,2,7,5]=>11
[1,3,4,6,5,2,7]=>4
[1,3,4,6,5,7,2]=>5
[1,3,4,6,7,2,5]=>12
[1,3,4,6,7,5,2]=>9
[1,3,4,7,2,5,6]=>10
[1,3,4,7,2,6,5]=>7
[1,3,4,7,5,2,6]=>7
[1,3,4,7,5,6,2]=>4
[1,3,4,7,6,2,5]=>11
[1,3,4,7,6,5,2]=>8
[1,3,5,2,4,6,7]=>5
[1,3,5,2,4,7,6]=>10
[1,3,5,2,6,4,7]=>8
[1,3,5,2,6,7,4]=>11
[1,3,5,2,7,4,6]=>13
[1,3,5,2,7,6,4]=>8
[1,3,5,4,2,6,7]=>3
[1,3,5,4,2,7,6]=>6
[1,3,5,4,6,2,7]=>4
[1,3,5,4,6,7,2]=>5
[1,3,5,4,7,2,6]=>7
[1,3,5,4,7,6,2]=>4
[1,3,5,6,2,4,7]=>9
[1,3,5,6,2,7,4]=>13
[1,3,5,6,4,2,7]=>7
[1,3,5,6,4,7,2]=>9
[1,3,5,6,7,2,4]=>14
[1,3,5,6,7,4,2]=>12
[1,3,5,7,2,4,6]=>14
[1,3,5,7,2,6,4]=>9
[1,3,5,7,4,2,6]=>12
[1,3,5,7,4,6,2]=>7
[1,3,5,7,6,2,4]=>13
[1,3,5,7,6,4,2]=>11
[1,3,6,2,4,5,7]=>7
[1,3,6,2,4,7,5]=>12
[1,3,6,2,5,4,7]=>5
[1,3,6,2,5,7,4]=>8
[1,3,6,2,7,4,5]=>15
[1,3,6,2,7,5,4]=>13
[1,3,6,4,2,5,7]=>5
[1,3,6,4,2,7,5]=>8
[1,3,6,4,5,2,7]=>3
[1,3,6,4,5,7,2]=>4
[1,3,6,4,7,2,5]=>9
[1,3,6,4,7,5,2]=>7
[1,3,6,5,2,4,7]=>8
[1,3,6,5,2,7,4]=>12
[1,3,6,5,4,2,7]=>6
[1,3,6,5,4,7,2]=>8
[1,3,6,5,7,2,4]=>13
[1,3,6,5,7,4,2]=>11
[1,3,6,7,2,4,5]=>16
[1,3,6,7,2,5,4]=>14
[1,3,6,7,4,2,5]=>14
[1,3,6,7,4,5,2]=>10
[1,3,6,7,5,2,4]=>9
[1,3,6,7,5,4,2]=>7
[1,3,7,2,4,5,6]=>9
[1,3,7,2,4,6,5]=>7
[1,3,7,2,5,4,6]=>7
[1,3,7,2,5,6,4]=>5
[1,3,7,2,6,4,5]=>12
[1,3,7,2,6,5,4]=>10
[1,3,7,4,2,5,6]=>7
[1,3,7,4,2,6,5]=>5
[1,3,7,4,5,2,6]=>5
[1,3,7,4,5,6,2]=>3
[1,3,7,4,6,2,5]=>8
[1,3,7,4,6,5,2]=>6
[1,3,7,5,2,4,6]=>12
[1,3,7,5,2,6,4]=>8
[1,3,7,5,4,2,6]=>10
[1,3,7,5,4,6,2]=>6
[1,3,7,5,6,2,4]=>11
[1,3,7,5,6,4,2]=>9
[1,3,7,6,2,4,5]=>15
[1,3,7,6,2,5,4]=>13
[1,3,7,6,4,2,5]=>13
[1,3,7,6,4,5,2]=>9
[1,3,7,6,5,2,4]=>8
[1,3,7,6,5,4,2]=>6
[1,4,2,3,5,6,7]=>3
[1,4,2,3,5,7,6]=>6
[1,4,2,3,6,5,7]=>6
[1,4,2,3,6,7,5]=>9
[1,4,2,3,7,5,6]=>9
[1,4,2,3,7,6,5]=>6
[1,4,2,5,3,6,7]=>5
[1,4,2,5,3,7,6]=>10
[1,4,2,5,6,3,7]=>7
[1,4,2,5,6,7,3]=>9
[1,4,2,5,7,3,6]=>12
[1,4,2,5,7,6,3]=>7
[1,4,2,6,3,5,7]=>8
[1,4,2,6,3,7,5]=>13
[1,4,2,6,5,3,7]=>5
[1,4,2,6,5,7,3]=>7
[1,4,2,6,7,3,5]=>15
[1,4,2,6,7,5,3]=>12
[1,4,2,7,3,5,6]=>11
[1,4,2,7,3,6,5]=>8
[1,4,2,7,5,3,6]=>8
[1,4,2,7,5,6,3]=>5
[1,4,2,7,6,3,5]=>13
[1,4,2,7,6,5,3]=>10
[1,4,3,2,5,6,7]=>2
[1,4,3,2,5,7,6]=>4
[1,4,3,2,6,5,7]=>4
[1,4,3,2,6,7,5]=>6
[1,4,3,2,7,5,6]=>6
[1,4,3,2,7,6,5]=>4
[1,4,3,5,2,6,7]=>3
[1,4,3,5,2,7,6]=>6
[1,4,3,5,6,2,7]=>4
[1,4,3,5,6,7,2]=>5
[1,4,3,5,7,2,6]=>7
[1,4,3,5,7,6,2]=>4
[1,4,3,6,2,5,7]=>5
[1,4,3,6,2,7,5]=>8
[1,4,3,6,5,2,7]=>3
[1,4,3,6,5,7,2]=>4
[1,4,3,6,7,2,5]=>9
[1,4,3,6,7,5,2]=>7
[1,4,3,7,2,5,6]=>7
[1,4,3,7,2,6,5]=>5
[1,4,3,7,5,2,6]=>5
[1,4,3,7,5,6,2]=>3
[1,4,3,7,6,2,5]=>8
[1,4,3,7,6,5,2]=>6
[1,4,5,2,3,6,7]=>6
[1,4,5,2,3,7,6]=>12
[1,4,5,2,6,3,7]=>9
[1,4,5,2,6,7,3]=>12
[1,4,5,2,7,3,6]=>15
[1,4,5,2,7,6,3]=>9
[1,4,5,3,2,6,7]=>5
[1,4,5,3,2,7,6]=>10
[1,4,5,3,6,2,7]=>7
[1,4,5,3,6,7,2]=>9
[1,4,5,3,7,2,6]=>12
[1,4,5,3,7,6,2]=>7
[1,4,5,6,2,3,7]=>10
[1,4,5,6,2,7,3]=>14
[1,4,5,6,3,2,7]=>9
[1,4,5,6,3,7,2]=>12
[1,4,5,6,7,2,3]=>15
[1,4,5,6,7,3,2]=>14
[1,4,5,7,2,3,6]=>16
[1,4,5,7,2,6,3]=>10
[1,4,5,7,3,2,6]=>15
[1,4,5,7,3,6,2]=>9
[1,4,5,7,6,2,3]=>14
[1,4,5,7,6,3,2]=>13
[1,4,6,2,3,5,7]=>9
[1,4,6,2,3,7,5]=>15
[1,4,6,2,5,3,7]=>6
[1,4,6,2,5,7,3]=>9
[1,4,6,2,7,3,5]=>18
[1,4,6,2,7,5,3]=>15
[1,4,6,3,2,5,7]=>8
[1,4,6,3,2,7,5]=>13
[1,4,6,3,5,2,7]=>5
[1,4,6,3,5,7,2]=>7
[1,4,6,3,7,2,5]=>15
[1,4,6,3,7,5,2]=>12
[1,4,6,5,2,3,7]=>9
[1,4,6,5,2,7,3]=>13
[1,4,6,5,3,2,7]=>8
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Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
References
[1] Ardila, F., Rincón, F., Williams, L. Positroids and non-crossing partitions arXiv:1308.2698
Code
def i_less(a, b, i):
"""
Return whether a < b for i < i+1 < ... < n < 1 < ... < i-1.
"""
if min(a, b) >= i or max(a, b) < i:
return a < b
return a >= i > b
def Grassmann_necklace(pi):
"""
Return the Grassmann necklace corresponding to pi.
sage: pi = Permutation([3,4,2,1])
sage: Grassmann_necklace(pi)
[{1, 2}, {2, 4}, {3, 4}, {1, 4}]
"""
pi = Permutation(pi)
N = []
n = len(pi)
for i in range(1,n+1):
iWEX = [j for j in range(1,n+1) if j == pi(j) or i_less(j, pi(j), i)]
N.append(set(iWEX))
return N
def Gale_less(S, T, i):
from functools import cmp_to_key
key = cmp_to_key(lambda a,b: 0 if a == b else int(-1) if i_less(a,b,i) else int(1))
S_sorted = sorted(S, key=key)
T_sorted = sorted(T, key=key)
return all(s == t or i_less(s, t, i) for s,t in zip(S_sorted, T_sorted))
def positroid(N):
"""
Return the positroid corresponding to the Grassmann necklace.
sage: pi = Permutation([3,4,2,1])
sage: N = Grassmann_necklace(pi)
sage: positroid(N)
[{1, 2}, {1, 3}, {1, 4}, {2, 4}, {3, 4}]
sage: [set([len(positroid(Grassmann_necklace(p))) for p in orbit(pi)]) for pi in Permutations(4)]
"""
n = len(N)
d = len(N[0])
result = []
for B in Subsets(range(1,n+1), d):
if all(B == N[j] or Gale_less(N[j], B, j+1) for j in range(n)):
result.append(B)
return result
def statistic(pi):
return len(positroid(Grassmann_necklace(pi)))
Created
Aug 12, 2019 at 16:34 by Martin Rubey
Updated
Mar 09, 2023 at 15:54 by Tilman Möller
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