Identifier
- St000339: Permutations ⟶ ℤ
Values
=>
[1]=>0
[1,2]=>0
[2,1]=>1
[1,2,3]=>0
[1,3,2]=>1
[2,1,3]=>3
[2,3,1]=>2
[3,1,2]=>1
[3,2,1]=>2
[1,2,3,4]=>0
[1,2,4,3]=>1
[1,3,2,4]=>3
[1,3,4,2]=>2
[1,4,2,3]=>1
[1,4,3,2]=>2
[2,1,3,4]=>5
[2,1,4,3]=>4
[2,3,1,4]=>5
[2,3,4,1]=>3
[2,4,1,3]=>2
[2,4,3,1]=>4
[3,1,2,4]=>4
[3,1,4,2]=>4
[3,2,1,4]=>4
[3,2,4,1]=>3
[3,4,1,2]=>2
[3,4,2,1]=>5
[4,1,2,3]=>1
[4,1,3,2]=>3
[4,2,1,3]=>2
[4,2,3,1]=>3
[4,3,1,2]=>3
[4,3,2,1]=>6
[1,2,3,4,5]=>0
[1,2,3,5,4]=>1
[1,2,4,3,5]=>3
[1,2,4,5,3]=>2
[1,2,5,3,4]=>1
[1,2,5,4,3]=>2
[1,3,2,4,5]=>5
[1,3,2,5,4]=>4
[1,3,4,2,5]=>5
[1,3,4,5,2]=>3
[1,3,5,2,4]=>2
[1,3,5,4,2]=>4
[1,4,2,3,5]=>4
[1,4,2,5,3]=>4
[1,4,3,2,5]=>4
[1,4,3,5,2]=>3
[1,4,5,2,3]=>2
[1,4,5,3,2]=>5
[1,5,2,3,4]=>1
[1,5,2,4,3]=>3
[1,5,3,2,4]=>2
[1,5,3,4,2]=>3
[1,5,4,2,3]=>3
[1,5,4,3,2]=>6
[2,1,3,4,5]=>7
[2,1,3,5,4]=>6
[2,1,4,3,5]=>8
[2,1,4,5,3]=>5
[2,1,5,3,4]=>4
[2,1,5,4,3]=>7
[2,3,1,4,5]=>8
[2,3,1,5,4]=>6
[2,3,4,1,5]=>7
[2,3,4,5,1]=>4
[2,3,5,1,4]=>3
[2,3,5,4,1]=>6
[2,4,1,3,5]=>6
[2,4,1,5,3]=>6
[2,4,3,1,5]=>7
[2,4,3,5,1]=>5
[2,4,5,1,3]=>3
[2,4,5,3,1]=>7
[2,5,1,3,4]=>2
[2,5,1,4,3]=>5
[2,5,3,1,4]=>4
[2,5,3,4,1]=>6
[2,5,4,1,3]=>5
[2,5,4,3,1]=>9
[3,1,2,4,5]=>7
[3,1,2,5,4]=>5
[3,1,4,2,5]=>8
[3,1,4,5,2]=>5
[3,1,5,2,4]=>4
[3,1,5,4,2]=>7
[3,2,1,4,5]=>6
[3,2,1,5,4]=>5
[3,2,4,1,5]=>6
[3,2,4,5,1]=>4
[3,2,5,1,4]=>3
[3,2,5,4,1]=>5
[3,4,1,2,5]=>6
[3,4,1,5,2]=>6
[3,4,2,1,5]=>9
[3,4,2,5,1]=>6
[3,4,5,1,2]=>3
[3,4,5,2,1]=>7
[3,5,1,2,4]=>2
[3,5,1,4,2]=>5
[3,5,2,1,4]=>5
[3,5,2,4,1]=>8
[3,5,4,1,2]=>5
[3,5,4,2,1]=>9
[4,1,2,3,5]=>5
[4,1,2,5,3]=>5
[4,1,3,2,5]=>6
[4,1,3,5,2]=>6
[4,1,5,2,3]=>4
[4,1,5,3,2]=>8
[4,2,1,3,5]=>5
[4,2,1,5,3]=>5
[4,2,3,1,5]=>5
[4,2,3,5,1]=>4
[4,2,5,1,3]=>3
[4,2,5,3,1]=>6
[4,3,1,2,5]=>7
[4,3,1,5,2]=>7
[4,3,2,1,5]=>10
[4,3,2,5,1]=>7
[4,3,5,1,2]=>4
[4,3,5,2,1]=>8
[4,5,1,2,3]=>2
[4,5,1,3,2]=>6
[4,5,2,1,3]=>5
[4,5,2,3,1]=>6
[4,5,3,1,2]=>4
[4,5,3,2,1]=>7
[5,1,2,3,4]=>1
[5,1,2,4,3]=>4
[5,1,3,2,4]=>3
[5,1,3,4,2]=>5
[5,1,4,2,3]=>4
[5,1,4,3,2]=>8
[5,2,1,3,4]=>2
[5,2,1,4,3]=>4
[5,2,3,1,4]=>3
[5,2,3,4,1]=>4
[5,2,4,1,3]=>4
[5,2,4,3,1]=>7
[5,3,1,2,4]=>3
[5,3,1,4,2]=>6
[5,3,2,1,4]=>6
[5,3,2,4,1]=>9
[5,3,4,1,2]=>4
[5,3,4,2,1]=>8
[5,4,1,2,3]=>3
[5,4,1,3,2]=>7
[5,4,2,1,3]=>6
[5,4,2,3,1]=>7
[5,4,3,1,2]=>5
[5,4,3,2,1]=>8
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>1
[1,2,3,5,4,6]=>3
[1,2,3,5,6,4]=>2
[1,2,3,6,4,5]=>1
[1,2,3,6,5,4]=>2
[1,2,4,3,5,6]=>5
[1,2,4,3,6,5]=>4
[1,2,4,5,3,6]=>5
[1,2,4,5,6,3]=>3
[1,2,4,6,3,5]=>2
[1,2,4,6,5,3]=>4
[1,2,5,3,4,6]=>4
[1,2,5,3,6,4]=>4
[1,2,5,4,3,6]=>4
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>2
[1,2,5,6,4,3]=>5
[1,2,6,3,4,5]=>1
[1,2,6,3,5,4]=>3
[1,2,6,4,3,5]=>2
[1,2,6,4,5,3]=>3
[1,2,6,5,3,4]=>3
[1,2,6,5,4,3]=>6
[1,3,2,4,5,6]=>7
[1,3,2,4,6,5]=>6
[1,3,2,5,4,6]=>8
[1,3,2,5,6,4]=>5
[1,3,2,6,4,5]=>4
[1,3,2,6,5,4]=>7
[1,3,4,2,5,6]=>8
[1,3,4,2,6,5]=>6
[1,3,4,5,2,6]=>7
[1,3,4,5,6,2]=>4
[1,3,4,6,2,5]=>3
[1,3,4,6,5,2]=>6
[1,3,5,2,4,6]=>6
[1,3,5,2,6,4]=>6
[1,3,5,4,2,6]=>7
[1,3,5,4,6,2]=>5
[1,3,5,6,2,4]=>3
[1,3,5,6,4,2]=>7
[1,3,6,2,4,5]=>2
[1,3,6,2,5,4]=>5
[1,3,6,4,2,5]=>4
[1,3,6,4,5,2]=>6
[1,3,6,5,2,4]=>5
[1,3,6,5,4,2]=>9
[1,4,2,3,5,6]=>7
[1,4,2,3,6,5]=>5
[1,4,2,5,3,6]=>8
[1,4,2,5,6,3]=>5
[1,4,2,6,3,5]=>4
[1,4,2,6,5,3]=>7
[1,4,3,2,5,6]=>6
[1,4,3,2,6,5]=>5
[1,4,3,5,2,6]=>6
[1,4,3,5,6,2]=>4
[1,4,3,6,2,5]=>3
[1,4,3,6,5,2]=>5
[1,4,5,2,3,6]=>6
[1,4,5,2,6,3]=>6
[1,4,5,3,2,6]=>9
[1,4,5,3,6,2]=>6
[1,4,5,6,2,3]=>3
[1,4,5,6,3,2]=>7
[1,4,6,2,3,5]=>2
[1,4,6,2,5,3]=>5
[1,4,6,3,2,5]=>5
[1,4,6,3,5,2]=>8
[1,4,6,5,2,3]=>5
[1,4,6,5,3,2]=>9
[1,5,2,3,4,6]=>5
[1,5,2,3,6,4]=>5
[1,5,2,4,3,6]=>6
[1,5,2,4,6,3]=>6
[1,5,2,6,3,4]=>4
[1,5,2,6,4,3]=>8
[1,5,3,2,4,6]=>5
[1,5,3,2,6,4]=>5
[1,5,3,4,2,6]=>5
[1,5,3,4,6,2]=>4
[1,5,3,6,2,4]=>3
[1,5,3,6,4,2]=>6
[1,5,4,2,3,6]=>7
[1,5,4,2,6,3]=>7
[1,5,4,3,2,6]=>10
[1,5,4,3,6,2]=>7
[1,5,4,6,2,3]=>4
[1,5,4,6,3,2]=>8
[1,5,6,2,3,4]=>2
[1,5,6,2,4,3]=>6
[1,5,6,3,2,4]=>5
[1,5,6,3,4,2]=>6
[1,5,6,4,2,3]=>4
[1,5,6,4,3,2]=>7
[1,6,2,3,4,5]=>1
[1,6,2,3,5,4]=>4
[1,6,2,4,3,5]=>3
[1,6,2,4,5,3]=>5
[1,6,2,5,3,4]=>4
[1,6,2,5,4,3]=>8
[1,6,3,2,4,5]=>2
[1,6,3,2,5,4]=>4
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>4
[1,6,3,5,2,4]=>4
[1,6,3,5,4,2]=>7
[1,6,4,2,3,5]=>3
[1,6,4,2,5,3]=>6
[1,6,4,3,2,5]=>6
[1,6,4,3,5,2]=>9
[1,6,4,5,2,3]=>4
[1,6,4,5,3,2]=>8
[1,6,5,2,3,4]=>3
[1,6,5,2,4,3]=>7
[1,6,5,3,2,4]=>6
[1,6,5,3,4,2]=>7
[1,6,5,4,2,3]=>5
[1,6,5,4,3,2]=>8
[2,1,3,4,5,6]=>9
[2,1,3,4,6,5]=>8
[2,1,3,5,4,6]=>10
[2,1,3,5,6,4]=>7
[2,1,3,6,4,5]=>6
[2,1,3,6,5,4]=>9
[2,1,4,3,5,6]=>12
[2,1,4,3,6,5]=>9
[2,1,4,5,3,6]=>10
[2,1,4,5,6,3]=>6
[2,1,4,6,3,5]=>5
[2,1,4,6,5,3]=>9
[2,1,5,3,4,6]=>9
[2,1,5,3,6,4]=>9
[2,1,5,4,3,6]=>11
[2,1,5,4,6,3]=>8
[2,1,5,6,3,4]=>5
[2,1,5,6,4,3]=>10
[2,1,6,3,4,5]=>4
[2,1,6,3,5,4]=>8
[2,1,6,4,3,5]=>7
[2,1,6,4,5,3]=>10
[2,1,6,5,3,4]=>8
[2,1,6,5,4,3]=>13
[2,3,1,4,5,6]=>11
[2,3,1,4,6,5]=>9
[2,3,1,5,4,6]=>11
[2,3,1,5,6,4]=>7
[2,3,1,6,4,5]=>6
[2,3,1,6,5,4]=>10
[2,3,4,1,5,6]=>11
[2,3,4,1,6,5]=>8
[2,3,4,5,1,6]=>9
[2,3,4,5,6,1]=>5
[2,3,4,6,1,5]=>4
[2,3,4,6,5,1]=>8
[2,3,5,1,4,6]=>8
[2,3,5,1,6,4]=>8
[2,3,5,4,1,6]=>10
[2,3,5,4,6,1]=>7
[2,3,5,6,1,4]=>4
[2,3,5,6,4,1]=>9
[2,3,6,1,4,5]=>3
[2,3,6,1,5,4]=>7
[2,3,6,4,1,5]=>6
[2,3,6,4,5,1]=>9
[2,3,6,5,1,4]=>7
[2,3,6,5,4,1]=>12
[2,4,1,3,5,6]=>10
[2,4,1,3,6,5]=>7
[2,4,1,5,3,6]=>11
[2,4,1,5,6,3]=>7
[2,4,1,6,3,5]=>6
[2,4,1,6,5,3]=>10
[2,4,3,1,5,6]=>10
[2,4,3,1,6,5]=>8
[2,4,3,5,1,6]=>9
[2,4,3,5,6,1]=>6
[2,4,3,6,1,5]=>5
[2,4,3,6,5,1]=>8
[2,4,5,1,3,6]=>8
[2,4,5,1,6,3]=>8
[2,4,5,3,1,6]=>12
[2,4,5,3,6,1]=>8
[2,4,5,6,1,3]=>4
[2,4,5,6,3,1]=>9
[2,4,6,1,3,5]=>3
[2,4,6,1,5,3]=>7
[2,4,6,3,1,5]=>7
[2,4,6,3,5,1]=>11
[2,4,6,5,1,3]=>7
[2,4,6,5,3,1]=>12
[2,5,1,3,4,6]=>7
[2,5,1,3,6,4]=>7
[2,5,1,4,3,6]=>9
[2,5,1,4,6,3]=>9
[2,5,1,6,3,4]=>6
[2,5,1,6,4,3]=>11
[2,5,3,1,4,6]=>8
[2,5,3,1,6,4]=>8
[2,5,3,4,1,6]=>9
[2,5,3,4,6,1]=>7
[2,5,3,6,1,4]=>5
[2,5,3,6,4,1]=>9
[2,5,4,1,3,6]=>10
[2,5,4,1,6,3]=>10
[2,5,4,3,1,6]=>14
[2,5,4,3,6,1]=>10
[2,5,4,6,1,3]=>6
[2,5,4,6,3,1]=>11
[2,5,6,1,3,4]=>3
[2,5,6,1,4,3]=>8
[2,5,6,3,1,4]=>7
[2,5,6,3,4,1]=>8
[2,5,6,4,1,3]=>6
[2,5,6,4,3,1]=>10
[2,6,1,3,4,5]=>2
[2,6,1,3,5,4]=>6
[2,6,1,4,3,5]=>5
[2,6,1,4,5,3]=>8
[2,6,1,5,3,4]=>6
[2,6,1,5,4,3]=>11
[2,6,3,1,4,5]=>4
[2,6,3,1,5,4]=>7
[2,6,3,4,1,5]=>6
[2,6,3,4,5,1]=>8
[2,6,3,5,1,4]=>7
[2,6,3,5,4,1]=>11
[2,6,4,1,3,5]=>5
[2,6,4,1,5,3]=>9
[2,6,4,3,1,5]=>9
[2,6,4,3,5,1]=>13
[2,6,4,5,1,3]=>6
[2,6,4,5,3,1]=>11
[2,6,5,1,3,4]=>5
[2,6,5,1,4,3]=>10
[2,6,5,3,1,4]=>9
[2,6,5,3,4,1]=>10
[2,6,5,4,1,3]=>8
[2,6,5,4,3,1]=>12
[3,1,2,4,5,6]=>10
[3,1,2,4,6,5]=>8
[3,1,2,5,4,6]=>10
[3,1,2,5,6,4]=>6
[3,1,2,6,4,5]=>5
[3,1,2,6,5,4]=>9
[3,1,4,2,5,6]=>12
[3,1,4,2,6,5]=>9
[3,1,4,5,2,6]=>10
[3,1,4,5,6,2]=>6
[3,1,4,6,2,5]=>5
[3,1,4,6,5,2]=>9
[3,1,5,2,4,6]=>9
[3,1,5,2,6,4]=>9
[3,1,5,4,2,6]=>11
[3,1,5,4,6,2]=>8
[3,1,5,6,2,4]=>5
[3,1,5,6,4,2]=>10
[3,1,6,2,4,5]=>4
[3,1,6,2,5,4]=>8
[3,1,6,4,2,5]=>7
[3,1,6,4,5,2]=>10
[3,1,6,5,2,4]=>8
[3,1,6,5,4,2]=>13
[3,2,1,4,5,6]=>8
[3,2,1,4,6,5]=>7
[3,2,1,5,4,6]=>9
[3,2,1,5,6,4]=>6
[3,2,1,6,4,5]=>5
[3,2,1,6,5,4]=>8
[3,2,4,1,5,6]=>9
[3,2,4,1,6,5]=>7
[3,2,4,5,1,6]=>8
[3,2,4,5,6,1]=>5
[3,2,4,6,1,5]=>4
[3,2,4,6,5,1]=>7
[3,2,5,1,4,6]=>7
[3,2,5,1,6,4]=>7
[3,2,5,4,1,6]=>8
[3,2,5,4,6,1]=>6
[3,2,5,6,1,4]=>4
[3,2,5,6,4,1]=>8
[3,2,6,1,4,5]=>3
[3,2,6,1,5,4]=>6
[3,2,6,4,1,5]=>5
[3,2,6,4,5,1]=>7
[3,2,6,5,1,4]=>6
[3,2,6,5,4,1]=>10
[3,4,1,2,5,6]=>10
[3,4,1,2,6,5]=>7
[3,4,1,5,2,6]=>11
[3,4,1,5,6,2]=>7
[3,4,1,6,2,5]=>6
[3,4,1,6,5,2]=>10
[3,4,2,1,5,6]=>13
[3,4,2,1,6,5]=>10
[3,4,2,5,1,6]=>11
[3,4,2,5,6,1]=>7
[3,4,2,6,1,5]=>6
[3,4,2,6,5,1]=>10
[3,4,5,1,2,6]=>8
[3,4,5,1,6,2]=>8
[3,4,5,2,1,6]=>12
[3,4,5,2,6,1]=>8
[3,4,5,6,1,2]=>4
[3,4,5,6,2,1]=>9
[3,4,6,1,2,5]=>3
[3,4,6,1,5,2]=>7
[3,4,6,2,1,5]=>7
[3,4,6,2,5,1]=>11
[3,4,6,5,1,2]=>7
[3,4,6,5,2,1]=>12
[3,5,1,2,4,6]=>7
[3,5,1,2,6,4]=>7
[3,5,1,4,2,6]=>9
[3,5,1,4,6,2]=>9
[3,5,1,6,2,4]=>6
[3,5,1,6,4,2]=>11
[3,5,2,1,4,6]=>10
[3,5,2,1,6,4]=>10
[3,5,2,4,1,6]=>12
[3,5,2,4,6,1]=>9
[3,5,2,6,1,4]=>6
[3,5,2,6,4,1]=>11
[3,5,4,1,2,6]=>10
[3,5,4,1,6,2]=>10
[3,5,4,2,1,6]=>14
[3,5,4,2,6,1]=>10
[3,5,4,6,1,2]=>6
[3,5,4,6,2,1]=>11
[3,5,6,1,2,4]=>3
[3,5,6,1,4,2]=>8
[3,5,6,2,1,4]=>7
[3,5,6,2,4,1]=>8
[3,5,6,4,1,2]=>6
[3,5,6,4,2,1]=>10
[3,6,1,2,4,5]=>2
[3,6,1,2,5,4]=>6
[3,6,1,4,2,5]=>5
[3,6,1,4,5,2]=>8
[3,6,1,5,2,4]=>6
[3,6,1,5,4,2]=>11
[3,6,2,1,4,5]=>5
[3,6,2,1,5,4]=>9
[3,6,2,4,1,5]=>8
[3,6,2,4,5,1]=>11
[3,6,2,5,1,4]=>6
[3,6,2,5,4,1]=>11
[3,6,4,1,2,5]=>5
[3,6,4,1,5,2]=>9
[3,6,4,2,1,5]=>9
[3,6,4,2,5,1]=>13
[3,6,4,5,1,2]=>6
[3,6,4,5,2,1]=>11
[3,6,5,1,2,4]=>5
[3,6,5,1,4,2]=>10
[3,6,5,2,1,4]=>9
[3,6,5,2,4,1]=>10
[3,6,5,4,1,2]=>8
[3,6,5,4,2,1]=>12
[4,1,2,3,5,6]=>9
[4,1,2,3,6,5]=>6
[4,1,2,5,3,6]=>10
[4,1,2,5,6,3]=>6
[4,1,2,6,3,5]=>5
[4,1,2,6,5,3]=>9
[4,1,3,2,5,6]=>9
[4,1,3,2,6,5]=>7
[4,1,3,5,2,6]=>10
[4,1,3,5,6,2]=>7
[4,1,3,6,2,5]=>6
[4,1,3,6,5,2]=>9
[4,1,5,2,3,6]=>9
[4,1,5,2,6,3]=>9
[4,1,5,3,2,6]=>13
[4,1,5,3,6,2]=>9
[4,1,5,6,2,3]=>5
[4,1,5,6,3,2]=>10
[4,1,6,2,3,5]=>4
[4,1,6,2,5,3]=>8
[4,1,6,3,2,5]=>8
[4,1,6,3,5,2]=>12
[4,1,6,5,2,3]=>8
[4,1,6,5,3,2]=>13
[4,2,1,3,5,6]=>8
[4,2,1,3,6,5]=>6
[4,2,1,5,3,6]=>9
[4,2,1,5,6,3]=>6
[4,2,1,6,3,5]=>5
[4,2,1,6,5,3]=>8
[4,2,3,1,5,6]=>7
[4,2,3,1,6,5]=>6
[4,2,3,5,1,6]=>7
[4,2,3,5,6,1]=>5
[4,2,3,6,1,5]=>4
[4,2,3,6,5,1]=>6
[4,2,5,1,3,6]=>7
[4,2,5,1,6,3]=>7
[4,2,5,3,1,6]=>10
[4,2,5,3,6,1]=>7
[4,2,5,6,1,3]=>4
[4,2,5,6,3,1]=>8
[4,2,6,1,3,5]=>3
[4,2,6,1,5,3]=>6
[4,2,6,3,1,5]=>6
[4,2,6,3,5,1]=>9
[4,2,6,5,1,3]=>6
[4,2,6,5,3,1]=>10
[4,3,1,2,5,6]=>11
[4,3,1,2,6,5]=>8
[4,3,1,5,2,6]=>12
[4,3,1,5,6,2]=>8
[4,3,1,6,2,5]=>7
[4,3,1,6,5,2]=>11
[4,3,2,1,5,6]=>14
[4,3,2,1,6,5]=>11
[4,3,2,5,1,6]=>12
[4,3,2,5,6,1]=>8
[4,3,2,6,1,5]=>7
[4,3,2,6,5,1]=>11
[4,3,5,1,2,6]=>9
[4,3,5,1,6,2]=>9
[4,3,5,2,1,6]=>13
[4,3,5,2,6,1]=>9
[4,3,5,6,1,2]=>5
[4,3,5,6,2,1]=>10
[4,3,6,1,2,5]=>4
[4,3,6,1,5,2]=>8
[4,3,6,2,1,5]=>8
[4,3,6,2,5,1]=>12
[4,3,6,5,1,2]=>8
[4,3,6,5,2,1]=>13
[4,5,1,2,3,6]=>7
[4,5,1,2,6,3]=>7
[4,5,1,3,2,6]=>11
[4,5,1,3,6,2]=>7
[4,5,1,6,2,3]=>6
[4,5,1,6,3,2]=>11
[4,5,2,1,3,6]=>10
[4,5,2,1,6,3]=>10
[4,5,2,3,1,6]=>11
[4,5,2,3,6,1]=>7
[4,5,2,6,1,3]=>6
[4,5,2,6,3,1]=>11
[4,5,3,1,2,6]=>8
[4,5,3,1,6,2]=>8
[4,5,3,2,1,6]=>11
[4,5,3,2,6,1]=>8
[4,5,3,6,1,2]=>5
[4,5,3,6,2,1]=>9
[4,5,6,1,2,3]=>3
[4,5,6,1,3,2]=>8
[4,5,6,2,1,3]=>7
[4,5,6,2,3,1]=>8
[4,5,6,3,1,2]=>7
[4,5,6,3,2,1]=>12
[4,6,1,2,3,5]=>2
[4,6,1,2,5,3]=>6
[4,6,1,3,2,5]=>6
[4,6,1,3,5,2]=>10
[4,6,1,5,2,3]=>6
[4,6,1,5,3,2]=>11
[4,6,2,1,3,5]=>5
[4,6,2,1,5,3]=>9
[4,6,2,3,1,5]=>6
[4,6,2,3,5,1]=>10
[4,6,2,5,1,3]=>6
[4,6,2,5,3,1]=>11
[4,6,3,1,2,5]=>4
[4,6,3,1,5,2]=>7
[4,6,3,2,1,5]=>7
[4,6,3,2,5,1]=>10
[4,6,3,5,1,2]=>7
[4,6,3,5,2,1]=>11
[4,6,5,1,2,3]=>5
[4,6,5,1,3,2]=>10
[4,6,5,2,1,3]=>9
[4,6,5,2,3,1]=>10
[4,6,5,3,1,2]=>9
[4,6,5,3,2,1]=>14
[5,1,2,3,4,6]=>6
[5,1,2,3,6,4]=>6
[5,1,2,4,3,6]=>8
[5,1,2,4,6,3]=>8
[5,1,2,6,3,4]=>5
[5,1,2,6,4,3]=>10
[5,1,3,2,4,6]=>7
[5,1,3,2,6,4]=>7
[5,1,3,4,2,6]=>8
[5,1,3,4,6,2]=>8
[5,1,3,6,2,4]=>6
[5,1,3,6,4,2]=>10
[5,1,4,2,3,6]=>9
[5,1,4,2,6,3]=>9
[5,1,4,3,2,6]=>13
[5,1,4,3,6,2]=>9
[5,1,4,6,2,3]=>5
[5,1,4,6,3,2]=>10
[5,1,6,2,3,4]=>4
[5,1,6,2,4,3]=>9
[5,1,6,3,2,4]=>8
[5,1,6,3,4,2]=>9
[5,1,6,4,2,3]=>7
[5,1,6,4,3,2]=>11
[5,2,1,3,4,6]=>6
[5,2,1,3,6,4]=>6
[5,2,1,4,3,6]=>7
[5,2,1,4,6,3]=>7
[5,2,1,6,3,4]=>5
[5,2,1,6,4,3]=>9
[5,2,3,1,4,6]=>6
[5,2,3,1,6,4]=>6
[5,2,3,4,1,6]=>6
[5,2,3,4,6,1]=>5
[5,2,3,6,1,4]=>4
[5,2,3,6,4,1]=>7
[5,2,4,1,3,6]=>8
[5,2,4,1,6,3]=>8
[5,2,4,3,1,6]=>11
[5,2,4,3,6,1]=>8
[5,2,4,6,1,3]=>5
[5,2,4,6,3,1]=>9
[5,2,6,1,3,4]=>3
[5,2,6,1,4,3]=>7
[5,2,6,3,1,4]=>6
[5,2,6,3,4,1]=>7
[5,2,6,4,1,3]=>5
[5,2,6,4,3,1]=>8
[5,3,1,2,4,6]=>8
[5,3,1,2,6,4]=>8
[5,3,1,4,2,6]=>10
[5,3,1,4,6,2]=>10
[5,3,1,6,2,4]=>7
[5,3,1,6,4,2]=>12
[5,3,2,1,4,6]=>11
[5,3,2,1,6,4]=>11
[5,3,2,4,1,6]=>13
[5,3,2,4,6,1]=>10
[5,3,2,6,1,4]=>7
[5,3,2,6,4,1]=>12
[5,3,4,1,2,6]=>9
[5,3,4,1,6,2]=>9
[5,3,4,2,1,6]=>13
[5,3,4,2,6,1]=>9
[5,3,4,6,1,2]=>5
[5,3,4,6,2,1]=>10
[5,3,6,1,2,4]=>4
[5,3,6,1,4,2]=>9
[5,3,6,2,1,4]=>8
[5,3,6,2,4,1]=>9
[5,3,6,4,1,2]=>7
[5,3,6,4,2,1]=>11
[5,4,1,2,3,6]=>8
[5,4,1,2,6,3]=>8
[5,4,1,3,2,6]=>12
[5,4,1,3,6,2]=>8
[5,4,1,6,2,3]=>7
[5,4,1,6,3,2]=>12
[5,4,2,1,3,6]=>11
[5,4,2,1,6,3]=>11
[5,4,2,3,1,6]=>12
[5,4,2,3,6,1]=>8
[5,4,2,6,1,3]=>7
[5,4,2,6,3,1]=>12
[5,4,3,1,2,6]=>9
[5,4,3,1,6,2]=>9
[5,4,3,2,1,6]=>12
[5,4,3,2,6,1]=>9
[5,4,3,6,1,2]=>6
[5,4,3,6,2,1]=>10
[5,4,6,1,2,3]=>4
[5,4,6,1,3,2]=>9
[5,4,6,2,1,3]=>8
[5,4,6,2,3,1]=>9
[5,4,6,3,1,2]=>8
[5,4,6,3,2,1]=>13
[5,6,1,2,3,4]=>2
[5,6,1,2,4,3]=>7
[5,6,1,3,2,4]=>6
[5,6,1,3,4,2]=>7
[5,6,1,4,2,3]=>5
[5,6,1,4,3,2]=>9
[5,6,2,1,3,4]=>5
[5,6,2,1,4,3]=>10
[5,6,2,3,1,4]=>6
[5,6,2,3,4,1]=>7
[5,6,2,4,1,3]=>8
[5,6,2,4,3,1]=>9
[5,6,3,1,2,4]=>4
[5,6,3,1,4,2]=>8
[5,6,3,2,1,4]=>7
[5,6,3,2,4,1]=>8
[5,6,3,4,1,2]=>6
[5,6,3,4,2,1]=>9
[5,6,4,1,2,3]=>5
[5,6,4,1,3,2]=>10
[5,6,4,2,1,3]=>9
[5,6,4,2,3,1]=>10
[5,6,4,3,1,2]=>9
[5,6,4,3,2,1]=>14
[6,1,2,3,4,5]=>1
[6,1,2,3,5,4]=>5
[6,1,2,4,3,5]=>4
[6,1,2,4,5,3]=>7
[6,1,2,5,3,4]=>5
[6,1,2,5,4,3]=>10
[6,1,3,2,4,5]=>3
[6,1,3,2,5,4]=>6
[6,1,3,4,2,5]=>5
[6,1,3,4,5,2]=>7
[6,1,3,5,2,4]=>6
[6,1,3,5,4,2]=>10
[6,1,4,2,3,5]=>4
[6,1,4,2,5,3]=>8
[6,1,4,3,2,5]=>8
[6,1,4,3,5,2]=>12
[6,1,4,5,2,3]=>5
[6,1,4,5,3,2]=>10
[6,1,5,2,3,4]=>4
[6,1,5,2,4,3]=>9
[6,1,5,3,2,4]=>8
[6,1,5,3,4,2]=>9
[6,1,5,4,2,3]=>7
[6,1,5,4,3,2]=>11
[6,2,1,3,4,5]=>2
[6,2,1,3,5,4]=>5
[6,2,1,4,3,5]=>4
[6,2,1,4,5,3]=>6
[6,2,1,5,3,4]=>5
[6,2,1,5,4,3]=>9
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>5
[6,2,3,4,1,5]=>4
[6,2,3,4,5,1]=>5
[6,2,3,5,1,4]=>5
[6,2,3,5,4,1]=>8
[6,2,4,1,3,5]=>4
[6,2,4,1,5,3]=>7
[6,2,4,3,1,5]=>7
[6,2,4,3,5,1]=>10
[6,2,4,5,1,3]=>5
[6,2,4,5,3,1]=>9
[6,2,5,1,3,4]=>4
[6,2,5,1,4,3]=>8
[6,2,5,3,1,4]=>7
[6,2,5,3,4,1]=>8
[6,2,5,4,1,3]=>6
[6,2,5,4,3,1]=>9
[6,3,1,2,4,5]=>3
[6,3,1,2,5,4]=>7
[6,3,1,4,2,5]=>6
[6,3,1,4,5,2]=>9
[6,3,1,5,2,4]=>7
[6,3,1,5,4,2]=>12
[6,3,2,1,4,5]=>6
[6,3,2,1,5,4]=>10
[6,3,2,4,1,5]=>9
[6,3,2,4,5,1]=>12
[6,3,2,5,1,4]=>7
[6,3,2,5,4,1]=>12
[6,3,4,1,2,5]=>4
[6,3,4,1,5,2]=>8
[6,3,4,2,1,5]=>8
[6,3,4,2,5,1]=>12
[6,3,4,5,1,2]=>5
[6,3,4,5,2,1]=>10
[6,3,5,1,2,4]=>4
[6,3,5,1,4,2]=>9
[6,3,5,2,1,4]=>8
[6,3,5,2,4,1]=>9
[6,3,5,4,1,2]=>7
[6,3,5,4,2,1]=>11
[6,4,1,2,3,5]=>3
[6,4,1,2,5,3]=>7
[6,4,1,3,2,5]=>7
[6,4,1,3,5,2]=>11
[6,4,1,5,2,3]=>7
[6,4,1,5,3,2]=>12
[6,4,2,1,3,5]=>6
[6,4,2,1,5,3]=>10
[6,4,2,3,1,5]=>7
[6,4,2,3,5,1]=>11
[6,4,2,5,1,3]=>7
[6,4,2,5,3,1]=>12
[6,4,3,1,2,5]=>5
[6,4,3,1,5,2]=>8
[6,4,3,2,1,5]=>8
[6,4,3,2,5,1]=>11
[6,4,3,5,1,2]=>6
[6,4,3,5,2,1]=>10
[6,4,5,1,2,3]=>4
[6,4,5,1,3,2]=>9
[6,4,5,2,1,3]=>8
[6,4,5,2,3,1]=>9
[6,4,5,3,1,2]=>8
[6,4,5,3,2,1]=>13
[6,5,1,2,3,4]=>3
[6,5,1,2,4,3]=>8
[6,5,1,3,2,4]=>7
[6,5,1,3,4,2]=>8
[6,5,1,4,2,3]=>6
[6,5,1,4,3,2]=>10
[6,5,2,1,3,4]=>6
[6,5,2,1,4,3]=>11
[6,5,2,3,1,4]=>7
[6,5,2,3,4,1]=>8
[6,5,2,4,1,3]=>9
[6,5,2,4,3,1]=>10
[6,5,3,1,2,4]=>5
[6,5,3,1,4,2]=>9
[6,5,3,2,1,4]=>8
[6,5,3,2,4,1]=>9
[6,5,3,4,1,2]=>7
[6,5,3,4,2,1]=>10
[6,5,4,1,2,3]=>6
[6,5,4,1,3,2]=>11
[6,5,4,2,1,3]=>10
[6,5,4,2,3,1]=>11
[6,5,4,3,1,2]=>10
[6,5,4,3,2,1]=>15
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Description
The maf index of a permutation.
Let $\sigma$ be a permutation with fixed point set $\operatorname{FIX}(\sigma)$, and let $\operatorname{Der}(\sigma)$ be the derangement obtained from $\sigma$ by removing the fixed points.
Then
$$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$$
where $\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of $\sigma$.
Let $\sigma$ be a permutation with fixed point set $\operatorname{FIX}(\sigma)$, and let $\operatorname{Der}(\sigma)$ be the derangement obtained from $\sigma$ by removing the fixed points.
Then
$$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$$
where $\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of $\sigma$.
References
[1] Foata, D., Han, G.-N. Fix-Mahonian Calculus, II: further statistics arXiv:math/0703101
[2] Han, G.-N., Xin, G. Permutations with Extremal number of Fixed Points arXiv:0706.1738
[2] Han, G.-N., Xin, G. Permutations with Extremal number of Fixed Points arXiv:0706.1738
Code
def statistic(x):
f = len(x.fixed_points())
return sum(x.fixed_points()) - ((f+1)*f)//2 + Der(x).major_index()
def Der(x):
return Word([i for i in sz(x) if i != 0]).standard_permutation()
#Returns the word (as a list) where fixed points of x are changed to zero (see [2])
def sz(x):
l = list(x)
for i in x.fixed_points():
l[i-1] = 0
return l
Created
Dec 18, 2015 at 13:45 by Joseph Bernstein
Updated
Dec 13, 2024 at 13:30 by Nupur Jain
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