Identifier
- St001079: Permutations ⟶ ℤ
Values
=>
[]=>0
[1]=>0
[1,2]=>0
[2,1]=>1
[1,2,3]=>0
[1,3,2]=>1
[2,1,3]=>1
[2,3,1]=>2
[3,1,2]=>2
[3,2,1]=>3
[1,2,3,4]=>0
[1,2,4,3]=>2
[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]=>1
[2,3,1,4]=>2
[2,3,4,1]=>4
[2,4,1,3]=>2
[2,4,3,1]=>4
[3,1,2,4]=>2
[3,1,4,2]=>2
[3,2,1,4]=>3
[3,2,4,1]=>3
[3,4,1,2]=>3
[3,4,2,1]=>4
[4,1,2,3]=>4
[4,1,3,2]=>4
[4,2,1,3]=>3
[4,2,3,1]=>3
[4,3,1,2]=>4
[4,3,2,1]=>4
[1,2,3,4,5]=>0
[1,2,3,5,4]=>6
[1,2,4,3,5]=>2
[1,2,4,5,3]=>6
[1,2,5,3,4]=>6
[1,2,5,4,3]=>7
[1,3,2,4,5]=>5
[1,3,2,5,4]=>1
[1,3,4,2,5]=>5
[1,3,4,5,2]=>6
[1,3,5,2,4]=>3
[1,3,5,4,2]=>7
[1,4,2,3,5]=>5
[1,4,2,5,3]=>3
[1,4,3,2,5]=>5
[1,4,3,5,2]=>6
[1,4,5,2,3]=>5
[1,4,5,3,2]=>8
[1,5,2,3,4]=>6
[1,5,2,4,3]=>7
[1,5,3,2,4]=>6
[1,5,3,4,2]=>4
[1,5,4,2,3]=>8
[1,5,4,3,2]=>6
[2,1,3,4,5]=>1
[2,1,3,5,4]=>5
[2,1,4,3,5]=>1
[2,1,4,5,3]=>7
[2,1,5,3,4]=>7
[2,1,5,4,3]=>7
[2,3,1,4,5]=>4
[2,3,1,5,4]=>2
[2,3,4,1,5]=>4
[2,3,4,5,1]=>7
[2,3,5,1,4]=>4
[2,3,5,4,1]=>8
[2,4,1,3,5]=>6
[2,4,1,5,3]=>2
[2,4,3,1,5]=>6
[2,4,3,5,1]=>5
[2,4,5,1,3]=>4
[2,4,5,3,1]=>7
[2,5,1,3,4]=>7
[2,5,1,4,3]=>7
[2,5,3,1,4]=>7
[2,5,3,4,1]=>5
[2,5,4,1,3]=>7
[2,5,4,3,1]=>5
[3,1,2,4,5]=>4
[3,1,2,5,4]=>2
[3,1,4,2,5]=>6
[3,1,4,5,2]=>7
[3,1,5,2,4]=>2
[3,1,5,4,2]=>7
[3,2,1,4,5]=>3
[3,2,1,5,4]=>3
[3,2,4,1,5]=>5
[3,2,4,5,1]=>8
[3,2,5,1,4]=>3
[3,2,5,4,1]=>8
[3,4,1,2,5]=>8
[3,4,1,5,2]=>7
[3,4,2,1,5]=>8
[3,4,2,5,1]=>6
[3,4,5,1,2]=>7
[3,4,5,2,1]=>6
[3,5,1,2,4]=>7
[3,5,1,4,2]=>3
[3,5,2,1,4]=>7
[3,5,2,4,1]=>4
[3,5,4,1,2]=>5
[3,5,4,2,1]=>6
[4,1,2,3,5]=>4
[4,1,2,5,3]=>4
[4,1,3,2,5]=>6
[4,1,3,5,2]=>7
[4,1,5,2,3]=>4
[4,1,5,3,2]=>7
[4,2,1,3,5]=>5
[4,2,1,5,3]=>3
[4,2,3,1,5]=>7
[4,2,3,5,1]=>6
[4,2,5,1,3]=>3
[4,2,5,3,1]=>6
[4,3,1,2,5]=>8
[4,3,1,5,2]=>7
[4,3,2,1,5]=>7
[4,3,2,5,1]=>7
[4,3,5,1,2]=>7
[4,3,5,2,1]=>7
[4,5,1,2,3]=>7
[4,5,1,3,2]=>5
[4,5,2,1,3]=>7
[4,5,2,3,1]=>4
[4,5,3,1,2]=>7
[4,5,3,2,1]=>6
[5,1,2,3,4]=>7
[5,1,2,4,3]=>8
[5,1,3,2,4]=>5
[5,1,3,4,2]=>5
[5,1,4,2,3]=>7
[5,1,4,3,2]=>5
[5,2,1,3,4]=>8
[5,2,1,4,3]=>8
[5,2,3,1,4]=>6
[5,2,3,4,1]=>6
[5,2,4,1,3]=>6
[5,2,4,3,1]=>6
[5,3,1,2,4]=>6
[5,3,1,4,2]=>4
[5,3,2,1,4]=>7
[5,3,2,4,1]=>5
[5,3,4,1,2]=>4
[5,3,4,2,1]=>5
[5,4,1,2,3]=>6
[5,4,1,3,2]=>6
[5,4,2,1,3]=>7
[5,4,2,3,1]=>5
[5,4,3,1,2]=>6
[5,4,3,2,1]=>5
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>11
[1,2,3,5,4,6]=>6
[1,2,3,5,6,4]=>11
[1,2,3,6,4,5]=>11
[1,2,3,6,5,4]=>9
[1,2,4,3,5,6]=>13
[1,2,4,3,6,5]=>2
[1,2,4,5,3,6]=>11
[1,2,4,5,6,3]=>9
[1,2,4,6,3,5]=>6
[1,2,4,6,5,3]=>13
[1,2,5,3,4,6]=>11
[1,2,5,3,6,4]=>6
[1,2,5,4,3,6]=>7
[1,2,5,4,6,3]=>11
[1,2,5,6,3,4]=>12
[1,2,5,6,4,3]=>16
[1,2,6,3,4,5]=>9
[1,2,6,3,5,4]=>13
[1,2,6,4,3,5]=>11
[1,2,6,4,5,3]=>8
[1,2,6,5,3,4]=>16
[1,2,6,5,4,3]=>12
[1,3,2,4,5,6]=>5
[1,3,2,4,6,5]=>10
[1,3,2,5,4,6]=>1
[1,3,2,5,6,4]=>10
[1,3,2,6,4,5]=>10
[1,3,2,6,5,4]=>12
[1,3,4,2,5,6]=>10
[1,3,4,2,6,5]=>5
[1,3,4,5,2,6]=>6
[1,3,4,5,6,2]=>10
[1,3,4,6,2,5]=>11
[1,3,4,6,5,2]=>15
[1,3,5,2,4,6]=>12
[1,3,5,2,6,4]=>3
[1,3,5,4,2,6]=>10
[1,3,5,4,6,2]=>8
[1,3,5,6,2,4]=>7
[1,3,5,6,4,2]=>12
[1,3,6,2,4,5]=>12
[1,3,6,2,5,4]=>10
[1,3,6,4,2,5]=>15
[1,3,6,4,5,2]=>11
[1,3,6,5,2,4]=>11
[1,3,6,5,4,2]=>9
[1,4,2,3,5,6]=>10
[1,4,2,3,6,5]=>5
[1,4,2,5,3,6]=>12
[1,4,2,5,6,3]=>12
[1,4,2,6,3,5]=>3
[1,4,2,6,5,3]=>10
[1,4,3,2,5,6]=>5
[1,4,3,2,6,5]=>10
[1,4,3,5,2,6]=>11
[1,4,3,5,6,2]=>15
[1,4,3,6,2,5]=>6
[1,4,3,6,5,2]=>10
[1,4,5,2,3,6]=>12
[1,4,5,2,6,3]=>12
[1,4,5,3,2,6]=>15
[1,4,5,3,6,2]=>11
[1,4,5,6,2,3]=>11
[1,4,5,6,3,2]=>11
[1,4,6,2,3,5]=>12
[1,4,6,2,5,3]=>5
[1,4,6,3,2,5]=>10
[1,4,6,3,5,2]=>8
[1,4,6,5,2,3]=>9
[1,4,6,5,3,2]=>12
[1,5,2,3,4,6]=>6
[1,5,2,3,6,4]=>11
[1,5,2,4,3,6]=>10
[1,5,2,4,6,3]=>15
[1,5,2,6,3,4]=>7
[1,5,2,6,4,3]=>11
[1,5,3,2,4,6]=>11
[1,5,3,2,6,4]=>6
[1,5,3,4,2,6]=>11
[1,5,3,4,6,2]=>11
[1,5,3,6,2,4]=>4
[1,5,3,6,4,2]=>9
[1,5,4,2,3,6]=>15
[1,5,4,2,6,3]=>10
[1,5,4,3,2,6]=>11
[1,5,4,3,6,2]=>11
[1,5,4,6,2,3]=>11
[1,5,4,6,3,2]=>12
[1,5,6,2,3,4]=>11
[1,5,6,2,4,3]=>9
[1,5,6,3,2,4]=>11
[1,5,6,3,4,2]=>6
[1,5,6,4,2,3]=>13
[1,5,6,4,3,2]=>10
[1,6,2,3,4,5]=>10
[1,6,2,3,5,4]=>15
[1,6,2,4,3,5]=>8
[1,6,2,4,5,3]=>11
[1,6,2,5,3,4]=>12
[1,6,2,5,4,3]=>9
[1,6,3,2,4,5]=>15
[1,6,3,2,5,4]=>10
[1,6,3,4,2,5]=>11
[1,6,3,4,5,2]=>10
[1,6,3,5,2,4]=>9
[1,6,3,5,4,2]=>12
[1,6,4,2,3,5]=>11
[1,6,4,2,5,3]=>8
[1,6,4,3,2,5]=>11
[1,6,4,3,5,2]=>11
[1,6,4,5,2,3]=>6
[1,6,4,5,3,2]=>9
[1,6,5,2,3,4]=>11
[1,6,5,2,4,3]=>12
[1,6,5,3,2,4]=>12
[1,6,5,3,4,2]=>9
[1,6,5,4,2,3]=>10
[1,6,5,4,3,2]=>7
[2,1,3,4,5,6]=>1
[2,1,3,4,6,5]=>12
[2,1,3,5,4,6]=>5
[2,1,3,5,6,4]=>12
[2,1,3,6,4,5]=>12
[2,1,3,6,5,4]=>10
[2,1,4,3,5,6]=>12
[2,1,4,3,6,5]=>1
[2,1,4,5,3,6]=>10
[2,1,4,5,6,3]=>8
[2,1,4,6,3,5]=>7
[2,1,4,6,5,3]=>12
[2,1,5,3,4,6]=>10
[2,1,5,3,6,4]=>7
[2,1,5,4,3,6]=>8
[2,1,5,4,6,3]=>12
[2,1,5,6,3,4]=>11
[2,1,5,6,4,3]=>15
[2,1,6,3,4,5]=>8
[2,1,6,3,5,4]=>12
[2,1,6,4,3,5]=>12
[2,1,6,4,5,3]=>7
[2,1,6,5,3,4]=>15
[2,1,6,5,4,3]=>13
[2,3,1,4,5,6]=>4
[2,3,1,4,6,5]=>11
[2,3,1,5,4,6]=>2
[2,3,1,5,6,4]=>11
[2,3,1,6,4,5]=>11
[2,3,1,6,5,4]=>13
[2,3,4,1,5,6]=>11
[2,3,4,1,6,5]=>4
[2,3,4,5,1,6]=>7
[2,3,4,5,6,1]=>9
[2,3,4,6,1,5]=>10
[2,3,4,6,5,1]=>14
[2,3,5,1,4,6]=>11
[2,3,5,1,6,4]=>4
[2,3,5,4,1,6]=>9
[2,3,5,4,6,1]=>9
[2,3,5,6,1,4]=>8
[2,3,5,6,4,1]=>13
[2,3,6,1,4,5]=>11
[2,3,6,1,5,4]=>11
[2,3,6,4,1,5]=>14
[2,3,6,4,5,1]=>10
[2,3,6,5,1,4]=>12
[2,3,6,5,4,1]=>10
[2,4,1,3,5,6]=>9
[2,4,1,3,6,5]=>6
[2,4,1,5,3,6]=>11
[2,4,1,5,6,3]=>13
[2,4,1,6,3,5]=>2
[2,4,1,6,5,3]=>9
[2,4,3,1,5,6]=>6
[2,4,3,1,6,5]=>9
[2,4,3,5,1,6]=>12
[2,4,3,5,6,1]=>14
[2,4,3,6,1,5]=>5
[2,4,3,6,5,1]=>10
[2,4,5,1,3,6]=>13
[2,4,5,1,6,3]=>11
[2,4,5,3,1,6]=>14
[2,4,5,3,6,1]=>12
[2,4,5,6,1,3]=>10
[2,4,5,6,3,1]=>10
[2,4,6,1,3,5]=>11
[2,4,6,1,5,3]=>4
[2,4,6,3,1,5]=>11
[2,4,6,3,5,1]=>7
[2,4,6,5,1,3]=>8
[2,4,6,5,3,1]=>11
[2,5,1,3,4,6]=>7
[2,5,1,3,6,4]=>10
[2,5,1,4,3,6]=>9
[2,5,1,4,6,3]=>14
[2,5,1,6,3,4]=>8
[2,5,1,6,4,3]=>12
[2,5,3,1,4,6]=>10
[2,5,3,1,6,4]=>7
[2,5,3,4,1,6]=>10
[2,5,3,4,6,1]=>12
[2,5,3,6,1,4]=>5
[2,5,3,6,4,1]=>10
[2,5,4,1,3,6]=>14
[2,5,4,1,6,3]=>9
[2,5,4,3,1,6]=>12
[2,5,4,3,6,1]=>10
[2,5,4,6,1,3]=>10
[2,5,4,6,3,1]=>13
[2,5,6,1,3,4]=>12
[2,5,6,1,4,3]=>10
[2,5,6,3,1,4]=>11
[2,5,6,3,4,1]=>7
[2,5,6,4,1,3]=>13
[2,5,6,4,3,1]=>10
[2,6,1,3,4,5]=>11
[2,6,1,3,5,4]=>14
[2,6,1,4,3,5]=>7
[2,6,1,4,5,3]=>12
[2,6,1,5,3,4]=>12
[2,6,1,5,4,3]=>8
[2,6,3,1,4,5]=>14
[2,6,3,1,5,4]=>11
[2,6,3,4,1,5]=>10
[2,6,3,4,5,1]=>11
[2,6,3,5,1,4]=>10
[2,6,3,5,4,1]=>11
[2,6,4,1,3,5]=>12
[2,6,4,1,5,3]=>7
[2,6,4,3,1,5]=>12
[2,6,4,3,5,1]=>10
[2,6,4,5,1,3]=>5
[2,6,4,5,3,1]=>8
[2,6,5,1,3,4]=>10
[2,6,5,1,4,3]=>12
[2,6,5,3,1,4]=>13
[2,6,5,3,4,1]=>10
[2,6,5,4,1,3]=>10
[2,6,5,4,3,1]=>7
[3,1,2,4,5,6]=>4
[3,1,2,4,6,5]=>11
[3,1,2,5,4,6]=>2
[3,1,2,5,6,4]=>11
[3,1,2,6,4,5]=>11
[3,1,2,6,5,4]=>13
[3,1,4,2,5,6]=>9
[3,1,4,2,6,5]=>6
[3,1,4,5,2,6]=>7
[3,1,4,5,6,2]=>11
[3,1,4,6,2,5]=>10
[3,1,4,6,5,2]=>14
[3,1,5,2,4,6]=>11
[3,1,5,2,6,4]=>2
[3,1,5,4,2,6]=>9
[3,1,5,4,6,2]=>7
[3,1,5,6,2,4]=>8
[3,1,5,6,4,2]=>12
[3,1,6,2,4,5]=>13
[3,1,6,2,5,4]=>9
[3,1,6,4,2,5]=>14
[3,1,6,4,5,2]=>12
[3,1,6,5,2,4]=>12
[3,1,6,5,4,2]=>8
[3,2,1,4,5,6]=>3
[3,2,1,4,6,5]=>12
[3,2,1,5,4,6]=>3
[3,2,1,5,6,4]=>12
[3,2,1,6,4,5]=>12
[3,2,1,6,5,4]=>12
[3,2,4,1,5,6]=>10
[3,2,4,1,6,5]=>5
[3,2,4,5,1,6]=>8
[3,2,4,5,6,1]=>10
[3,2,4,6,1,5]=>9
[3,2,4,6,5,1]=>14
[3,2,5,1,4,6]=>12
[3,2,5,1,6,4]=>3
[3,2,5,4,1,6]=>8
[3,2,5,4,6,1]=>8
[3,2,5,6,1,4]=>9
[3,2,5,6,4,1]=>13
[3,2,6,1,4,5]=>12
[3,2,6,1,5,4]=>10
[3,2,6,4,1,5]=>13
[3,2,6,4,5,1]=>11
[3,2,6,5,1,4]=>13
[3,2,6,5,4,1]=>9
[3,4,1,2,5,6]=>8
[3,4,1,2,6,5]=>9
[3,4,1,5,2,6]=>8
[3,4,1,5,6,2]=>13
[3,4,1,6,2,5]=>7
[3,4,1,6,5,2]=>12
[3,4,2,1,5,6]=>9
[3,4,2,1,6,5]=>8
[3,4,2,5,1,6]=>9
[3,4,2,5,6,1]=>13
[3,4,2,6,1,5]=>6
[3,4,2,6,5,1]=>11
[3,4,5,1,2,6]=>13
[3,4,5,1,6,2]=>8
[3,4,5,2,1,6]=>13
[3,4,5,2,6,1]=>9
[3,4,5,6,1,2]=>11
[3,4,5,6,2,1]=>12
[3,4,6,1,2,5]=>13
[3,4,6,1,5,2]=>9
[3,4,6,2,1,5]=>12
[3,4,6,2,5,1]=>8
[3,4,6,5,1,2]=>12
[3,4,6,5,2,1]=>11
[3,5,1,2,4,6]=>8
[3,5,1,2,6,4]=>7
[3,5,1,4,2,6]=>10
[3,5,1,4,6,2]=>12
[3,5,1,6,2,4]=>3
[3,5,1,6,4,2]=>8
[3,5,2,1,4,6]=>7
[3,5,2,1,6,4]=>8
[3,5,2,4,1,6]=>11
[3,5,2,4,6,1]=>13
[3,5,2,6,1,4]=>4
[3,5,2,6,4,1]=>9
[3,5,4,1,2,6]=>12
[3,5,4,1,6,2]=>10
[3,5,4,2,1,6]=>13
[3,5,4,2,6,1]=>11
[3,5,4,6,1,2]=>9
[3,5,4,6,2,1]=>10
[3,5,6,1,2,4]=>10
[3,5,6,1,4,2]=>5
[3,5,6,2,1,4]=>11
[3,5,6,2,4,1]=>6
[3,5,6,4,1,2]=>9
[3,5,6,4,2,1]=>10
[3,6,1,2,4,5]=>13
[3,6,1,2,5,4]=>12
[3,6,1,4,2,5]=>12
[3,6,1,4,5,2]=>9
[3,6,1,5,2,4]=>8
[3,6,1,5,4,2]=>11
[3,6,2,1,4,5]=>12
[3,6,2,1,5,4]=>13
[3,6,2,4,1,5]=>11
[3,6,2,4,5,1]=>10
[3,6,2,5,1,4]=>9
[3,6,2,5,4,1]=>12
[3,6,4,1,2,5]=>11
[3,6,4,1,5,2]=>12
[3,6,4,2,1,5]=>12
[3,6,4,2,5,1]=>11
[3,6,4,5,1,2]=>9
[3,6,4,5,2,1]=>8
[3,6,5,1,2,4]=>11
[3,6,5,1,4,2]=>8
[3,6,5,2,1,4]=>11
[3,6,5,2,4,1]=>9
[3,6,5,4,1,2]=>6
[3,6,5,4,2,1]=>7
[4,1,2,3,5,6]=>11
[4,1,2,3,6,5]=>4
[4,1,2,5,3,6]=>11
[4,1,2,5,6,3]=>11
[4,1,2,6,3,5]=>4
[4,1,2,6,5,3]=>11
[4,1,3,2,5,6]=>6
[4,1,3,2,6,5]=>9
[4,1,3,5,2,6]=>10
[4,1,3,5,6,2]=>14
[4,1,3,6,2,5]=>7
[4,1,3,6,5,2]=>11
[4,1,5,2,3,6]=>13
[4,1,5,2,6,3]=>11
[4,1,5,3,2,6]=>14
[4,1,5,3,6,2]=>12
[4,1,5,6,2,3]=>12
[4,1,5,6,3,2]=>10
[4,1,6,2,3,5]=>11
[4,1,6,2,5,3]=>4
[4,1,6,3,2,5]=>9
[4,1,6,3,5,2]=>7
[4,1,6,5,2,3]=>10
[4,1,6,5,3,2]=>12
[4,2,1,3,5,6]=>10
[4,2,1,3,6,5]=>5
[4,2,1,5,3,6]=>12
[4,2,1,5,6,3]=>12
[4,2,1,6,3,5]=>3
[4,2,1,6,5,3]=>10
[4,2,3,1,5,6]=>7
[4,2,3,1,6,5]=>8
[4,2,3,5,1,6]=>11
[4,2,3,5,6,1]=>13
[4,2,3,6,1,5]=>6
[4,2,3,6,5,1]=>11
[4,2,5,1,3,6]=>13
[4,2,5,1,6,3]=>10
[4,2,5,3,1,6]=>14
[4,2,5,3,6,1]=>11
[4,2,5,6,1,3]=>11
[4,2,5,6,3,1]=>9
[4,2,6,1,3,5]=>10
[4,2,6,1,5,3]=>3
[4,2,6,3,1,5]=>10
[4,2,6,3,5,1]=>6
[4,2,6,5,1,3]=>9
[4,2,6,5,3,1]=>11
[4,3,1,2,5,6]=>9
[4,3,1,2,6,5]=>8
[4,3,1,5,2,6]=>7
[4,3,1,5,6,2]=>12
[4,3,1,6,2,5]=>8
[4,3,1,6,5,2]=>13
[4,3,2,1,5,6]=>10
[4,3,2,1,6,5]=>7
[4,3,2,5,1,6]=>8
[4,3,2,5,6,1]=>12
[4,3,2,6,1,5]=>7
[4,3,2,6,5,1]=>12
[4,3,5,1,2,6]=>13
[4,3,5,1,6,2]=>9
[4,3,5,2,1,6]=>14
[4,3,5,2,6,1]=>10
[4,3,5,6,1,2]=>12
[4,3,5,6,2,1]=>13
[4,3,6,1,2,5]=>12
[4,3,6,1,5,2]=>8
[4,3,6,2,1,5]=>12
[4,3,6,2,5,1]=>7
[4,3,6,5,1,2]=>11
[4,3,6,5,2,1]=>10
[4,5,1,2,3,6]=>13
[4,5,1,2,6,3]=>13
[4,5,1,3,2,6]=>12
[4,5,1,3,6,2]=>11
[4,5,1,6,2,3]=>10
[4,5,1,6,3,2]=>11
[4,5,2,1,3,6]=>13
[4,5,2,1,6,3]=>12
[4,5,2,3,1,6]=>12
[4,5,2,3,6,1]=>10
[4,5,2,6,1,3]=>9
[4,5,2,6,3,1]=>10
[4,5,3,1,2,6]=>13
[4,5,3,1,6,2]=>12
[4,5,3,2,1,6]=>12
[4,5,3,2,6,1]=>12
[4,5,3,6,1,2]=>9
[4,5,3,6,2,1]=>10
[4,5,6,1,2,3]=>10
[4,5,6,1,3,2]=>10
[4,5,6,2,1,3]=>10
[4,5,6,2,3,1]=>9
[4,5,6,3,1,2]=>8
[4,5,6,3,2,1]=>7
[4,6,1,2,3,5]=>8
[4,6,1,2,5,3]=>9
[4,6,1,3,2,5]=>10
[4,6,1,3,5,2]=>12
[4,6,1,5,2,3]=>5
[4,6,1,5,3,2]=>8
[4,6,2,1,3,5]=>9
[4,6,2,1,5,3]=>8
[4,6,2,3,1,5]=>9
[4,6,2,3,5,1]=>11
[4,6,2,5,1,3]=>4
[4,6,2,5,3,1]=>7
[4,6,3,1,2,5]=>12
[4,6,3,1,5,2]=>10
[4,6,3,2,1,5]=>11
[4,6,3,2,5,1]=>9
[4,6,3,5,1,2]=>9
[4,6,3,5,2,1]=>8
[4,6,5,1,2,3]=>10
[4,6,5,1,3,2]=>7
[4,6,5,2,1,3]=>9
[4,6,5,2,3,1]=>6
[4,6,5,3,1,2]=>10
[4,6,5,3,2,1]=>10
[5,1,2,3,4,6]=>7
[5,1,2,3,6,4]=>10
[5,1,2,4,3,6]=>9
[5,1,2,4,6,3]=>14
[5,1,2,6,3,4]=>8
[5,1,2,6,4,3]=>12
[5,1,3,2,4,6]=>12
[5,1,3,2,6,4]=>5
[5,1,3,4,2,6]=>10
[5,1,3,4,6,2]=>10
[5,1,3,6,2,4]=>5
[5,1,3,6,4,2]=>10
[5,1,4,2,3,6]=>14
[5,1,4,2,6,3]=>11
[5,1,4,3,2,6]=>12
[5,1,4,3,6,2]=>12
[5,1,4,6,2,3]=>11
[5,1,4,6,3,2]=>13
[5,1,6,2,3,4]=>10
[5,1,6,2,4,3]=>8
[5,1,6,3,2,4]=>10
[5,1,6,3,4,2]=>5
[5,1,6,4,2,3]=>13
[5,1,6,4,3,2]=>10
[5,2,1,3,4,6]=>8
[5,2,1,3,6,4]=>9
[5,2,1,4,3,6]=>8
[5,2,1,4,6,3]=>13
[5,2,1,6,3,4]=>9
[5,2,1,6,4,3]=>13
[5,2,3,1,4,6]=>11
[5,2,3,1,6,4]=>6
[5,2,3,4,1,6]=>9
[5,2,3,4,6,1]=>11
[5,2,3,6,1,4]=>6
[5,2,3,6,4,1]=>11
[5,2,4,1,3,6]=>14
[5,2,4,1,6,3]=>10
[5,2,4,3,1,6]=>13
[5,2,4,3,6,1]=>11
[5,2,4,6,1,3]=>11
[5,2,4,6,3,1]=>12
[5,2,6,1,3,4]=>11
[5,2,6,1,4,3]=>9
[5,2,6,3,1,4]=>11
[5,2,6,3,4,1]=>6
[5,2,6,4,1,3]=>12
[5,2,6,4,3,1]=>9
[5,3,1,2,4,6]=>9
[5,3,1,2,6,4]=>6
[5,3,1,4,2,6]=>11
[5,3,1,4,6,2]=>11
[5,3,1,6,2,4]=>4
[5,3,1,6,4,2]=>9
[5,3,2,1,4,6]=>8
[5,3,2,1,6,4]=>7
[5,3,2,4,1,6]=>12
[5,3,2,4,6,1]=>12
[5,3,2,6,1,4]=>5
[5,3,2,6,4,1]=>10
[5,3,4,1,2,6]=>12
[5,3,4,1,6,2]=>9
[5,3,4,2,1,6]=>13
[5,3,4,2,6,1]=>10
[5,3,4,6,1,2]=>10
[5,3,4,6,2,1]=>10
[5,3,6,1,2,4]=>9
[5,3,6,1,4,2]=>4
[5,3,6,2,1,4]=>10
[5,3,6,2,4,1]=>5
[5,3,6,4,1,2]=>10
[5,3,6,4,2,1]=>10
[5,4,1,2,3,6]=>13
[5,4,1,2,6,3]=>12
[5,4,1,3,2,6]=>13
[5,4,1,3,6,2]=>12
[5,4,1,6,2,3]=>11
[5,4,1,6,3,2]=>11
[5,4,2,1,3,6]=>14
[5,4,2,1,6,3]=>12
[5,4,2,3,1,6]=>13
[5,4,2,3,6,1]=>11
[5,4,2,6,1,3]=>10
[5,4,2,6,3,1]=>11
[5,4,3,1,2,6]=>12
[5,4,3,1,6,2]=>11
[5,4,3,2,1,6]=>11
[5,4,3,2,6,1]=>11
[5,4,3,6,1,2]=>8
[5,4,3,6,2,1]=>9
[5,4,6,1,2,3]=>10
[5,4,6,1,3,2]=>9
[5,4,6,2,1,3]=>9
[5,4,6,2,3,1]=>8
[5,4,6,3,1,2]=>9
[5,4,6,3,2,1]=>8
[5,6,1,2,3,4]=>11
[5,6,1,2,4,3]=>12
[5,6,1,3,2,4]=>9
[5,6,1,3,4,2]=>9
[5,6,1,4,2,3]=>9
[5,6,1,4,3,2]=>6
[5,6,2,1,3,4]=>12
[5,6,2,1,4,3]=>11
[5,6,2,3,1,4]=>10
[5,6,2,3,4,1]=>8
[5,6,2,4,1,3]=>10
[5,6,2,4,3,1]=>7
[5,6,3,1,2,4]=>9
[5,6,3,1,4,2]=>9
[5,6,3,2,1,4]=>8
[5,6,3,2,4,1]=>10
[5,6,3,4,1,2]=>5
[5,6,3,4,2,1]=>6
[5,6,4,1,2,3]=>8
[5,6,4,1,3,2]=>10
[5,6,4,2,1,3]=>9
[5,6,4,2,3,1]=>9
[5,6,4,3,1,2]=>8
[5,6,4,3,2,1]=>7
[6,1,2,3,4,5]=>9
[6,1,2,3,5,4]=>14
[6,1,2,4,3,5]=>9
[6,1,2,4,5,3]=>10
[6,1,2,5,3,4]=>13
[6,1,2,5,4,3]=>10
[6,1,3,2,4,5]=>14
[6,1,3,2,5,4]=>10
[6,1,3,4,2,5]=>12
[6,1,3,4,5,2]=>11
[6,1,3,5,2,4]=>10
[6,1,3,5,4,2]=>11
[6,1,4,2,3,5]=>12
[6,1,4,2,5,3]=>7
[6,1,4,3,2,5]=>10
[6,1,4,3,5,2]=>10
[6,1,4,5,2,3]=>7
[6,1,4,5,3,2]=>10
[6,1,5,2,3,4]=>10
[6,1,5,2,4,3]=>11
[6,1,5,3,2,4]=>13
[6,1,5,3,4,2]=>8
[6,1,5,4,2,3]=>10
[6,1,5,4,3,2]=>7
[6,2,1,3,4,5]=>10
[6,2,1,3,5,4]=>14
[6,2,1,4,3,5]=>8
[6,2,1,4,5,3]=>11
[6,2,1,5,3,4]=>13
[6,2,1,5,4,3]=>9
[6,2,3,1,4,5]=>13
[6,2,3,1,5,4]=>11
[6,2,3,4,1,5]=>11
[6,2,3,4,5,1]=>12
[6,2,3,5,1,4]=>11
[6,2,3,5,4,1]=>12
[6,2,4,1,3,5]=>11
[6,2,4,1,5,3]=>6
[6,2,4,3,1,5]=>11
[6,2,4,3,5,1]=>9
[6,2,4,5,1,3]=>6
[6,2,4,5,3,1]=>9
[6,2,5,1,3,4]=>9
[6,2,5,1,4,3]=>11
[6,2,5,3,1,4]=>12
[6,2,5,3,4,1]=>9
[6,2,5,4,1,3]=>9
[6,2,5,4,3,1]=>6
[6,3,1,2,4,5]=>13
[6,3,1,2,5,4]=>11
[6,3,1,4,2,5]=>13
[6,3,1,4,5,2]=>10
[6,3,1,5,2,4]=>9
[6,3,1,5,4,2]=>12
[6,3,2,1,4,5]=>12
[6,3,2,1,5,4]=>12
[6,3,2,4,1,5]=>12
[6,3,2,4,5,1]=>11
[6,3,2,5,1,4]=>10
[6,3,2,5,4,1]=>12
[6,3,4,1,2,5]=>10
[6,3,4,1,5,2]=>11
[6,3,4,2,1,5]=>11
[6,3,4,2,5,1]=>10
[6,3,4,5,1,2]=>8
[6,3,4,5,2,1]=>7
[6,3,5,1,2,4]=>10
[6,3,5,1,4,2]=>7
[6,3,5,2,1,4]=>11
[6,3,5,2,4,1]=>8
[6,3,5,4,1,2]=>7
[6,3,5,4,2,1]=>8
[6,4,1,2,3,5]=>9
[6,4,1,2,5,3]=>8
[6,4,1,3,2,5]=>11
[6,4,1,3,5,2]=>11
[6,4,1,5,2,3]=>6
[6,4,1,5,3,2]=>9
[6,4,2,1,3,5]=>10
[6,4,2,1,5,3]=>7
[6,4,2,3,1,5]=>10
[6,4,2,3,5,1]=>10
[6,4,2,5,1,3]=>5
[6,4,2,5,3,1]=>8
[6,4,3,1,2,5]=>12
[6,4,3,1,5,2]=>9
[6,4,3,2,1,5]=>11
[6,4,3,2,5,1]=>8
[6,4,3,5,1,2]=>10
[6,4,3,5,2,1]=>9
[6,4,5,1,2,3]=>9
[6,4,5,1,3,2]=>6
[6,4,5,2,1,3]=>8
[6,4,5,2,3,1]=>5
[6,4,5,3,1,2]=>9
[6,4,5,3,2,1]=>10
[6,5,1,2,3,4]=>12
[6,5,1,2,4,3]=>11
[6,5,1,3,2,4]=>10
[6,5,1,3,4,2]=>8
[6,5,1,4,2,3]=>10
[6,5,1,4,3,2]=>7
[6,5,2,1,3,4]=>13
[6,5,2,1,4,3]=>10
[6,5,2,3,1,4]=>10
[6,5,2,3,4,1]=>7
[6,5,2,4,1,3]=>10
[6,5,2,4,3,1]=>8
[6,5,3,1,2,4]=>10
[6,5,3,1,4,2]=>8
[6,5,3,2,1,4]=>9
[6,5,3,2,4,1]=>9
[6,5,3,4,1,2]=>6
[6,5,3,4,2,1]=>7
[6,5,4,1,2,3]=>7
[6,5,4,1,3,2]=>10
[6,5,4,2,1,3]=>8
[6,5,4,2,3,1]=>10
[6,5,4,3,1,2]=>7
[6,5,4,3,2,1]=>6
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Description
The minimal length of a factorization of a permutation using the permutations (12)(34)..., (23)(45)..., and (12).
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k} \},$$
where, with $m_1$ the largest even number at most $n$ and $m_2$ the largest odd number at most $n$, each factor $\tau_i$ is one of the three permutations $(1,2)(3,4)\cdots(m_1-1,m_1)$ or $(2,3)(4,5)\cdots(m_2-1,m_2)$ or $(1,2)$.
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k} \},$$
where, with $m_1$ the largest even number at most $n$ and $m_2$ the largest odd number at most $n$, each factor $\tau_i$ is one of the three permutations $(1,2)(3,4)\cdots(m_1-1,m_1)$ or $(2,3)(4,5)\cdots(m_2-1,m_2)$ or $(1,2)$.
References
[1] Pak, I. "Natural" generating sets for symmetric groups. MathOverflow:24128
Code
def statistic(pi):
def gens(n):
n1 = n if is_even(n) else n-1
n2 = n-2 if is_even(n) else n-1
return [Permutation([i+2 if is_even(i) else i for i in range(n1)]),
Permutation([1] + [i+3 if is_even(i) else i+1 for i in range(n2)]), Permutation([2,1])]
G = PermutationGroup(gens(len(pi)))
pi = G(pi)
w = gap.Factorization(G._gap_(), pi._gap_())
return w.Length()
Created
Jan 08, 2018 at 23:38 by Martin Rubey
Updated
Jan 09, 2018 at 19:40 by Martin Rubey
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