Identifier
- St000677: Permutations ⟶ ℤ
Values
=>
[1,2]=>0
[2,1]=>1
[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]=>1
[1,2,4,3]=>2
[1,3,2,4]=>2
[1,3,4,2]=>1
[1,4,2,3]=>1
[1,4,3,2]=>2
[2,1,3,4]=>2
[2,1,4,3]=>3
[2,3,1,4]=>1
[2,3,4,1]=>2
[2,4,1,3]=>0
[2,4,3,1]=>3
[3,1,2,4]=>1
[3,1,4,2]=>4
[3,2,1,4]=>2
[3,2,4,1]=>3
[3,4,1,2]=>1
[3,4,2,1]=>2
[4,1,2,3]=>2
[4,1,3,2]=>3
[4,2,1,3]=>3
[4,2,3,1]=>2
[4,3,1,2]=>2
[4,3,2,1]=>3
[1,2,3,4,5]=>1
[1,2,3,5,4]=>2
[1,2,4,3,5]=>2
[1,2,4,5,3]=>1
[1,2,5,3,4]=>1
[1,2,5,4,3]=>2
[1,3,2,4,5]=>2
[1,3,2,5,4]=>3
[1,3,4,2,5]=>1
[1,3,4,5,2]=>2
[1,3,5,2,4]=>0
[1,3,5,4,2]=>3
[1,4,2,3,5]=>1
[1,4,2,5,3]=>4
[1,4,3,2,5]=>2
[1,4,3,5,2]=>3
[1,4,5,2,3]=>1
[1,4,5,3,2]=>2
[1,5,2,3,4]=>2
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>2
[1,5,4,2,3]=>2
[1,5,4,3,2]=>3
[2,1,3,4,5]=>2
[2,1,3,5,4]=>3
[2,1,4,3,5]=>3
[2,1,4,5,3]=>2
[2,1,5,3,4]=>2
[2,1,5,4,3]=>3
[2,3,1,4,5]=>1
[2,3,1,5,4]=>2
[2,3,4,1,5]=>2
[2,3,4,5,1]=>1
[2,3,5,1,4]=>1
[2,3,5,4,1]=>2
[2,4,1,3,5]=>0
[2,4,1,5,3]=>3
[2,4,3,1,5]=>3
[2,4,3,5,1]=>2
[2,4,5,1,3]=>2
[2,4,5,3,1]=>1
[2,5,1,3,4]=>1
[2,5,1,4,3]=>2
[2,5,3,1,4]=>4
[2,5,3,4,1]=>1
[2,5,4,1,3]=>3
[2,5,4,3,1]=>2
[3,1,2,4,5]=>1
[3,1,2,5,4]=>2
[3,1,4,2,5]=>4
[3,1,4,5,2]=>1
[3,1,5,2,4]=>3
[3,1,5,4,2]=>2
[3,2,1,4,5]=>2
[3,2,1,5,4]=>3
[3,2,4,1,5]=>3
[3,2,4,5,1]=>2
[3,2,5,1,4]=>2
[3,2,5,4,1]=>3
[3,4,1,2,5]=>1
[3,4,1,5,2]=>2
[3,4,2,1,5]=>2
[3,4,2,5,1]=>1
[3,4,5,1,2]=>1
[3,4,5,2,1]=>2
[3,5,1,2,4]=>2
[3,5,1,4,2]=>1
[3,5,2,1,4]=>3
[3,5,2,4,1]=>0
[3,5,4,1,2]=>2
[3,5,4,2,1]=>3
[4,1,2,3,5]=>2
[4,1,2,5,3]=>1
[4,1,3,2,5]=>3
[4,1,3,5,2]=>0
[4,1,5,2,3]=>2
[4,1,5,3,2]=>3
[4,2,1,3,5]=>3
[4,2,1,5,3]=>2
[4,2,3,1,5]=>2
[4,2,3,5,1]=>1
[4,2,5,1,3]=>1
[4,2,5,3,1]=>4
[4,3,1,2,5]=>2
[4,3,1,5,2]=>3
[4,3,2,1,5]=>3
[4,3,2,5,1]=>2
[4,3,5,1,2]=>2
[4,3,5,2,1]=>3
[4,5,1,2,3]=>1
[4,5,1,3,2]=>2
[4,5,2,1,3]=>2
[4,5,2,3,1]=>1
[4,5,3,1,2]=>1
[4,5,3,2,1]=>2
[5,1,2,3,4]=>1
[5,1,2,4,3]=>2
[5,1,3,2,4]=>2
[5,1,3,4,2]=>1
[5,1,4,2,3]=>1
[5,1,4,3,2]=>2
[5,2,1,3,4]=>2
[5,2,1,4,3]=>3
[5,2,3,1,4]=>1
[5,2,3,4,1]=>2
[5,2,4,1,3]=>0
[5,2,4,3,1]=>3
[5,3,1,2,4]=>1
[5,3,1,4,2]=>4
[5,3,2,1,4]=>2
[5,3,2,4,1]=>3
[5,3,4,1,2]=>1
[5,3,4,2,1]=>2
[5,4,1,2,3]=>2
[5,4,1,3,2]=>3
[5,4,2,1,3]=>3
[5,4,2,3,1]=>2
[5,4,3,1,2]=>2
[5,4,3,2,1]=>3
[1,2,3,4,5,6]=>3
[1,2,3,4,6,5]=>4
[1,2,3,5,4,6]=>4
[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]=>4
[1,2,4,3,6,5]=>5
[1,2,4,5,3,6]=>3
[1,2,4,5,6,3]=>4
[1,2,4,6,3,5]=>2
[1,2,4,6,5,3]=>5
[1,2,5,3,4,6]=>3
[1,2,5,3,6,4]=>6
[1,2,5,4,3,6]=>4
[1,2,5,4,6,3]=>5
[1,2,5,6,3,4]=>3
[1,2,5,6,4,3]=>4
[1,2,6,3,4,5]=>4
[1,2,6,3,5,4]=>5
[1,2,6,4,3,5]=>5
[1,2,6,4,5,3]=>4
[1,2,6,5,3,4]=>4
[1,2,6,5,4,3]=>5
[1,3,2,4,5,6]=>4
[1,3,2,4,6,5]=>5
[1,3,2,5,4,6]=>5
[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]=>4
[1,3,4,5,6,2]=>3
[1,3,4,6,2,5]=>3
[1,3,4,6,5,2]=>4
[1,3,5,2,4,6]=>2
[1,3,5,2,6,4]=>5
[1,3,5,4,2,6]=>5
[1,3,5,4,6,2]=>4
[1,3,5,6,2,4]=>4
[1,3,5,6,4,2]=>3
[1,3,6,2,4,5]=>3
[1,3,6,2,5,4]=>4
[1,3,6,4,2,5]=>6
[1,3,6,4,5,2]=>3
[1,3,6,5,2,4]=>5
[1,3,6,5,4,2]=>4
[1,4,2,3,5,6]=>3
[1,4,2,3,6,5]=>4
[1,4,2,5,3,6]=>6
[1,4,2,5,6,3]=>3
[1,4,2,6,3,5]=>5
[1,4,2,6,5,3]=>4
[1,4,3,2,5,6]=>4
[1,4,3,2,6,5]=>5
[1,4,3,5,2,6]=>5
[1,4,3,5,6,2]=>4
[1,4,3,6,2,5]=>4
[1,4,3,6,5,2]=>5
[1,4,5,2,3,6]=>3
[1,4,5,2,6,3]=>4
[1,4,5,3,2,6]=>4
[1,4,5,3,6,2]=>3
[1,4,5,6,2,3]=>3
[1,4,5,6,3,2]=>4
[1,4,6,2,3,5]=>4
[1,4,6,2,5,3]=>3
[1,4,6,3,2,5]=>5
[1,4,6,3,5,2]=>2
[1,4,6,5,2,3]=>4
[1,4,6,5,3,2]=>5
[1,5,2,3,4,6]=>4
[1,5,2,3,6,4]=>3
[1,5,2,4,3,6]=>5
[1,5,2,4,6,3]=>2
[1,5,2,6,3,4]=>4
[1,5,2,6,4,3]=>5
[1,5,3,2,4,6]=>5
[1,5,3,2,6,4]=>4
[1,5,3,4,2,6]=>4
[1,5,3,4,6,2]=>3
[1,5,3,6,2,4]=>3
[1,5,3,6,4,2]=>6
[1,5,4,2,3,6]=>4
[1,5,4,2,6,3]=>5
[1,5,4,3,2,6]=>5
[1,5,4,3,6,2]=>4
[1,5,4,6,2,3]=>4
[1,5,4,6,3,2]=>5
[1,5,6,2,3,4]=>3
[1,5,6,2,4,3]=>4
[1,5,6,3,2,4]=>4
[1,5,6,3,4,2]=>3
[1,5,6,4,2,3]=>3
[1,5,6,4,3,2]=>4
[1,6,2,3,4,5]=>3
[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]=>3
[1,6,2,5,4,3]=>4
[1,6,3,2,4,5]=>4
[1,6,3,2,5,4]=>5
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>4
[1,6,3,5,2,4]=>2
[1,6,3,5,4,2]=>5
[1,6,4,2,3,5]=>3
[1,6,4,2,5,3]=>6
[1,6,4,3,2,5]=>4
[1,6,4,3,5,2]=>5
[1,6,4,5,2,3]=>3
[1,6,4,5,3,2]=>4
[1,6,5,2,3,4]=>4
[1,6,5,2,4,3]=>5
[1,6,5,3,2,4]=>5
[1,6,5,3,4,2]=>4
[1,6,5,4,2,3]=>4
[1,6,5,4,3,2]=>5
[2,1,3,4,5,6]=>4
[2,1,3,4,6,5]=>5
[2,1,3,5,4,6]=>5
[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]=>5
[2,1,4,3,6,5]=>6
[2,1,4,5,3,6]=>4
[2,1,4,5,6,3]=>5
[2,1,4,6,3,5]=>3
[2,1,4,6,5,3]=>6
[2,1,5,3,4,6]=>4
[2,1,5,3,6,4]=>7
[2,1,5,4,3,6]=>5
[2,1,5,4,6,3]=>6
[2,1,5,6,3,4]=>4
[2,1,5,6,4,3]=>5
[2,1,6,3,4,5]=>5
[2,1,6,3,5,4]=>6
[2,1,6,4,3,5]=>6
[2,1,6,4,5,3]=>5
[2,1,6,5,3,4]=>5
[2,1,6,5,4,3]=>6
[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]=>3
[2,3,1,6,4,5]=>3
[2,3,1,6,5,4]=>4
[2,3,4,1,5,6]=>4
[2,3,4,1,6,5]=>5
[2,3,4,5,1,6]=>3
[2,3,4,5,6,1]=>4
[2,3,4,6,1,5]=>2
[2,3,4,6,5,1]=>5
[2,3,5,1,4,6]=>3
[2,3,5,1,6,4]=>6
[2,3,5,4,1,6]=>4
[2,3,5,4,6,1]=>5
[2,3,5,6,1,4]=>3
[2,3,5,6,4,1]=>4
[2,3,6,1,4,5]=>4
[2,3,6,1,5,4]=>5
[2,3,6,4,1,5]=>5
[2,3,6,4,5,1]=>4
[2,3,6,5,1,4]=>4
[2,3,6,5,4,1]=>5
[2,4,1,3,5,6]=>2
[2,4,1,3,6,5]=>3
[2,4,1,5,3,6]=>5
[2,4,1,5,6,3]=>2
[2,4,1,6,3,5]=>4
[2,4,1,6,5,3]=>3
[2,4,3,1,5,6]=>5
[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]=>3
[2,4,3,6,5,1]=>6
[2,4,5,1,3,6]=>4
[2,4,5,1,6,3]=>5
[2,4,5,3,1,6]=>3
[2,4,5,3,6,1]=>4
[2,4,5,6,1,3]=>2
[2,4,5,6,3,1]=>5
[2,4,6,1,3,5]=>5
[2,4,6,1,5,3]=>4
[2,4,6,3,1,5]=>4
[2,4,6,3,5,1]=>3
[2,4,6,5,1,3]=>3
[2,4,6,5,3,1]=>6
[2,5,1,3,4,6]=>3
[2,5,1,3,6,4]=>2
[2,5,1,4,3,6]=>4
[2,5,1,4,6,3]=>1
[2,5,1,6,3,4]=>3
[2,5,1,6,4,3]=>4
[2,5,3,1,4,6]=>6
[2,5,3,1,6,4]=>5
[2,5,3,4,1,6]=>3
[2,5,3,4,6,1]=>4
[2,5,3,6,1,4]=>2
[2,5,3,6,4,1]=>7
[2,5,4,1,3,6]=>5
[2,5,4,1,6,3]=>6
[2,5,4,3,1,6]=>4
[2,5,4,3,6,1]=>5
[2,5,4,6,1,3]=>3
[2,5,4,6,3,1]=>6
[2,5,6,1,3,4]=>4
[2,5,6,1,4,3]=>5
[2,5,6,3,1,4]=>3
[2,5,6,3,4,1]=>4
[2,5,6,4,1,3]=>2
[2,5,6,4,3,1]=>5
[2,6,1,3,4,5]=>2
[2,6,1,3,5,4]=>3
[2,6,1,4,3,5]=>3
[2,6,1,4,5,3]=>2
[2,6,1,5,3,4]=>2
[2,6,1,5,4,3]=>3
[2,6,3,1,4,5]=>5
[2,6,3,1,5,4]=>6
[2,6,3,4,1,5]=>2
[2,6,3,4,5,1]=>5
[2,6,3,5,1,4]=>1
[2,6,3,5,4,1]=>6
[2,6,4,1,3,5]=>4
[2,6,4,1,5,3]=>7
[2,6,4,3,1,5]=>3
[2,6,4,3,5,1]=>6
[2,6,4,5,1,3]=>2
[2,6,4,5,3,1]=>5
[2,6,5,1,3,4]=>5
[2,6,5,1,4,3]=>6
[2,6,5,3,1,4]=>4
[2,6,5,3,4,1]=>5
[2,6,5,4,1,3]=>3
[2,6,5,4,3,1]=>6
[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]=>3
[3,1,2,6,4,5]=>3
[3,1,2,6,5,4]=>4
[3,1,4,2,5,6]=>6
[3,1,4,2,6,5]=>7
[3,1,4,5,2,6]=>3
[3,1,4,5,6,2]=>6
[3,1,4,6,2,5]=>2
[3,1,4,6,5,2]=>7
[3,1,5,2,4,6]=>5
[3,1,5,2,6,4]=>8
[3,1,5,4,2,6]=>4
[3,1,5,4,6,2]=>7
[3,1,5,6,2,4]=>3
[3,1,5,6,4,2]=>6
[3,1,6,2,4,5]=>6
[3,1,6,2,5,4]=>7
[3,1,6,4,2,5]=>5
[3,1,6,4,5,2]=>6
[3,1,6,5,2,4]=>4
[3,1,6,5,4,2]=>7
[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]=>4
[3,2,1,6,4,5]=>4
[3,2,1,6,5,4]=>5
[3,2,4,1,5,6]=>5
[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]=>3
[3,2,4,6,5,1]=>6
[3,2,5,1,4,6]=>4
[3,2,5,1,6,4]=>7
[3,2,5,4,1,6]=>5
[3,2,5,4,6,1]=>6
[3,2,5,6,1,4]=>4
[3,2,5,6,4,1]=>5
[3,2,6,1,4,5]=>5
[3,2,6,1,5,4]=>6
[3,2,6,4,1,5]=>6
[3,2,6,4,5,1]=>5
[3,2,6,5,1,4]=>5
[3,2,6,5,4,1]=>6
[3,4,1,2,5,6]=>3
[3,4,1,2,6,5]=>4
[3,4,1,5,2,6]=>4
[3,4,1,5,6,2]=>3
[3,4,1,6,2,5]=>3
[3,4,1,6,5,2]=>4
[3,4,2,1,5,6]=>4
[3,4,2,1,6,5]=>5
[3,4,2,5,1,6]=>3
[3,4,2,5,6,1]=>4
[3,4,2,6,1,5]=>2
[3,4,2,6,5,1]=>5
[3,4,5,1,2,6]=>3
[3,4,5,1,6,2]=>6
[3,4,5,2,1,6]=>4
[3,4,5,2,6,1]=>5
[3,4,5,6,1,2]=>3
[3,4,5,6,2,1]=>4
[3,4,6,1,2,5]=>4
[3,4,6,1,5,2]=>5
[3,4,6,2,1,5]=>5
[3,4,6,2,5,1]=>4
[3,4,6,5,1,2]=>4
[3,4,6,5,2,1]=>5
[3,5,1,2,4,6]=>4
[3,5,1,2,6,4]=>3
[3,5,1,4,2,6]=>3
[3,5,1,4,6,2]=>2
[3,5,1,6,2,4]=>2
[3,5,1,6,4,2]=>5
[3,5,2,1,4,6]=>5
[3,5,2,1,6,4]=>4
[3,5,2,4,1,6]=>2
[3,5,2,4,6,1]=>3
[3,5,2,6,1,4]=>1
[3,5,2,6,4,1]=>6
[3,5,4,1,2,6]=>4
[3,5,4,1,6,2]=>7
[3,5,4,2,1,6]=>5
[3,5,4,2,6,1]=>6
[3,5,4,6,1,2]=>4
[3,5,4,6,2,1]=>5
[3,5,6,1,2,4]=>3
[3,5,6,1,4,2]=>6
[3,5,6,2,1,4]=>4
[3,5,6,2,4,1]=>5
[3,5,6,4,1,2]=>3
[3,5,6,4,2,1]=>4
[3,6,1,2,4,5]=>3
[3,6,1,2,5,4]=>4
[3,6,1,4,2,5]=>2
[3,6,1,4,5,2]=>3
[3,6,1,5,2,4]=>1
[3,6,1,5,4,2]=>4
[3,6,2,1,4,5]=>4
[3,6,2,1,5,4]=>5
[3,6,2,4,1,5]=>1
[3,6,2,4,5,1]=>4
[3,6,2,5,1,4]=>0
[3,6,2,5,4,1]=>5
[3,6,4,1,2,5]=>3
[3,6,4,1,5,2]=>8
[3,6,4,2,1,5]=>4
[3,6,4,2,5,1]=>7
[3,6,4,5,1,2]=>3
[3,6,4,5,2,1]=>4
[3,6,5,1,2,4]=>4
[3,6,5,1,4,2]=>7
[3,6,5,2,1,4]=>5
[3,6,5,2,4,1]=>6
[3,6,5,4,1,2]=>4
[3,6,5,4,2,1]=>5
[4,1,2,3,5,6]=>4
[4,1,2,3,6,5]=>5
[4,1,2,5,3,6]=>3
[4,1,2,5,6,3]=>4
[4,1,2,6,3,5]=>2
[4,1,2,6,5,3]=>5
[4,1,3,2,5,6]=>5
[4,1,3,2,6,5]=>6
[4,1,3,5,2,6]=>2
[4,1,3,5,6,2]=>5
[4,1,3,6,2,5]=>1
[4,1,3,6,5,2]=>6
[4,1,5,2,3,6]=>4
[4,1,5,2,6,3]=>9
[4,1,5,3,2,6]=>5
[4,1,5,3,6,2]=>8
[4,1,5,6,2,3]=>4
[4,1,5,6,3,2]=>5
[4,1,6,2,3,5]=>5
[4,1,6,2,5,3]=>8
[4,1,6,3,2,5]=>6
[4,1,6,3,5,2]=>7
[4,1,6,5,2,3]=>5
[4,1,6,5,3,2]=>6
[4,2,1,3,5,6]=>5
[4,2,1,3,6,5]=>6
[4,2,1,5,3,6]=>4
[4,2,1,5,6,3]=>5
[4,2,1,6,3,5]=>3
[4,2,1,6,5,3]=>6
[4,2,3,1,5,6]=>4
[4,2,3,1,6,5]=>5
[4,2,3,5,1,6]=>3
[4,2,3,5,6,1]=>4
[4,2,3,6,1,5]=>2
[4,2,3,6,5,1]=>5
[4,2,5,1,3,6]=>3
[4,2,5,1,6,3]=>8
[4,2,5,3,1,6]=>6
[4,2,5,3,6,1]=>7
[4,2,5,6,1,3]=>5
[4,2,5,6,3,1]=>4
[4,2,6,1,3,5]=>4
[4,2,6,1,5,3]=>7
[4,2,6,3,1,5]=>7
[4,2,6,3,5,1]=>6
[4,2,6,5,1,3]=>6
[4,2,6,5,3,1]=>5
[4,3,1,2,5,6]=>4
[4,3,1,2,6,5]=>5
[4,3,1,5,2,6]=>5
[4,3,1,5,6,2]=>4
[4,3,1,6,2,5]=>4
[4,3,1,6,5,2]=>5
[4,3,2,1,5,6]=>5
[4,3,2,1,6,5]=>6
[4,3,2,5,1,6]=>4
[4,3,2,5,6,1]=>5
[4,3,2,6,1,5]=>3
[4,3,2,6,5,1]=>6
[4,3,5,1,2,6]=>4
[4,3,5,1,6,2]=>7
[4,3,5,2,1,6]=>5
[4,3,5,2,6,1]=>6
[4,3,5,6,1,2]=>4
[4,3,5,6,2,1]=>5
[4,3,6,1,2,5]=>5
[4,3,6,1,5,2]=>6
[4,3,6,2,1,5]=>6
[4,3,6,2,5,1]=>5
[4,3,6,5,1,2]=>5
[4,3,6,5,2,1]=>6
[4,5,1,2,3,6]=>3
[4,5,1,2,6,3]=>4
[4,5,1,3,2,6]=>4
[4,5,1,3,6,2]=>3
[4,5,1,6,2,3]=>3
[4,5,1,6,3,2]=>4
[4,5,2,1,3,6]=>4
[4,5,2,1,6,3]=>5
[4,5,2,3,1,6]=>3
[4,5,2,3,6,1]=>4
[4,5,2,6,1,3]=>2
[4,5,2,6,3,1]=>5
[4,5,3,1,2,6]=>3
[4,5,3,1,6,2]=>6
[4,5,3,2,1,6]=>4
[4,5,3,2,6,1]=>5
[4,5,3,6,1,2]=>3
[4,5,3,6,2,1]=>4
[4,5,6,1,2,3]=>4
[4,5,6,1,3,2]=>5
[4,5,6,2,1,3]=>5
[4,5,6,2,3,1]=>4
[4,5,6,3,1,2]=>4
[4,5,6,3,2,1]=>5
[4,6,1,2,3,5]=>2
[4,6,1,2,5,3]=>5
[4,6,1,3,2,5]=>3
[4,6,1,3,5,2]=>4
[4,6,1,5,2,3]=>2
[4,6,1,5,3,2]=>3
[4,6,2,1,3,5]=>3
[4,6,2,1,5,3]=>6
[4,6,2,3,1,5]=>2
[4,6,2,3,5,1]=>5
[4,6,2,5,1,3]=>1
[4,6,2,5,3,1]=>4
[4,6,3,1,2,5]=>2
[4,6,3,1,5,2]=>7
[4,6,3,2,1,5]=>3
[4,6,3,2,5,1]=>6
[4,6,3,5,1,2]=>2
[4,6,3,5,2,1]=>3
[4,6,5,1,2,3]=>5
[4,6,5,1,3,2]=>6
[4,6,5,2,1,3]=>6
[4,6,5,2,3,1]=>5
[4,6,5,3,1,2]=>5
[4,6,5,3,2,1]=>6
[5,1,2,3,4,6]=>3
[5,1,2,3,6,4]=>6
[5,1,2,4,3,6]=>4
[5,1,2,4,6,3]=>5
[5,1,2,6,3,4]=>3
[5,1,2,6,4,3]=>4
[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]=>6
[5,1,3,6,2,4]=>2
[5,1,3,6,4,2]=>5
[5,1,4,2,3,6]=>3
[5,1,4,2,6,3]=>8
[5,1,4,3,2,6]=>4
[5,1,4,3,6,2]=>7
[5,1,4,6,2,3]=>3
[5,1,4,6,3,2]=>4
[5,1,6,2,3,4]=>6
[5,1,6,2,4,3]=>7
[5,1,6,3,2,4]=>7
[5,1,6,3,4,2]=>6
[5,1,6,4,2,3]=>6
[5,1,6,4,3,2]=>7
[5,2,1,3,4,6]=>4
[5,2,1,3,6,4]=>7
[5,2,1,4,3,6]=>5
[5,2,1,4,6,3]=>6
[5,2,1,6,3,4]=>4
[5,2,1,6,4,3]=>5
[5,2,3,1,4,6]=>3
[5,2,3,1,6,4]=>6
[5,2,3,4,1,6]=>4
[5,2,3,4,6,1]=>5
[5,2,3,6,1,4]=>3
[5,2,3,6,4,1]=>4
[5,2,4,1,3,6]=>2
[5,2,4,1,6,3]=>7
[5,2,4,3,1,6]=>5
[5,2,4,3,6,1]=>6
[5,2,4,6,1,3]=>4
[5,2,4,6,3,1]=>3
[5,2,6,1,3,4]=>5
[5,2,6,1,4,3]=>6
[5,2,6,3,1,4]=>8
[5,2,6,3,4,1]=>5
[5,2,6,4,1,3]=>7
[5,2,6,4,3,1]=>6
[5,3,1,2,4,6]=>3
[5,3,1,2,6,4]=>6
[5,3,1,4,2,6]=>6
[5,3,1,4,6,2]=>5
[5,3,1,6,2,4]=>5
[5,3,1,6,4,2]=>4
[5,3,2,1,4,6]=>4
[5,3,2,1,6,4]=>7
[5,3,2,4,1,6]=>5
[5,3,2,4,6,1]=>6
[5,3,2,6,1,4]=>4
[5,3,2,6,4,1]=>5
[5,3,4,1,2,6]=>3
[5,3,4,1,6,2]=>6
[5,3,4,2,1,6]=>4
[5,3,4,2,6,1]=>5
[5,3,4,6,1,2]=>3
[5,3,4,6,2,1]=>4
[5,3,6,1,2,4]=>6
[5,3,6,1,4,2]=>5
[5,3,6,2,1,4]=>7
[5,3,6,2,4,1]=>4
[5,3,6,4,1,2]=>6
[5,3,6,4,2,1]=>7
[5,4,1,2,3,6]=>4
[5,4,1,2,6,3]=>5
[5,4,1,3,2,6]=>5
[5,4,1,3,6,2]=>4
[5,4,1,6,2,3]=>4
[5,4,1,6,3,2]=>5
[5,4,2,1,3,6]=>5
[5,4,2,1,6,3]=>6
[5,4,2,3,1,6]=>4
[5,4,2,3,6,1]=>5
[5,4,2,6,1,3]=>3
[5,4,2,6,3,1]=>6
[5,4,3,1,2,6]=>4
[5,4,3,1,6,2]=>7
[5,4,3,2,1,6]=>5
[5,4,3,2,6,1]=>6
[5,4,3,6,1,2]=>4
[5,4,3,6,2,1]=>5
[5,4,6,1,2,3]=>5
[5,4,6,1,3,2]=>6
[5,4,6,2,1,3]=>6
[5,4,6,2,3,1]=>5
[5,4,6,3,1,2]=>5
[5,4,6,3,2,1]=>6
[5,6,1,2,3,4]=>3
[5,6,1,2,4,3]=>4
[5,6,1,3,2,4]=>4
[5,6,1,3,4,2]=>3
[5,6,1,4,2,3]=>3
[5,6,1,4,3,2]=>4
[5,6,2,1,3,4]=>4
[5,6,2,1,4,3]=>5
[5,6,2,3,1,4]=>3
[5,6,2,3,4,1]=>4
[5,6,2,4,1,3]=>2
[5,6,2,4,3,1]=>5
[5,6,3,1,2,4]=>3
[5,6,3,1,4,2]=>6
[5,6,3,2,1,4]=>4
[5,6,3,2,4,1]=>5
[5,6,3,4,1,2]=>3
[5,6,3,4,2,1]=>4
[5,6,4,1,2,3]=>4
[5,6,4,1,3,2]=>5
[5,6,4,2,1,3]=>5
[5,6,4,2,3,1]=>4
[5,6,4,3,1,2]=>4
[5,6,4,3,2,1]=>5
[6,1,2,3,4,5]=>4
[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]=>4
[6,1,2,5,4,3]=>5
[6,1,3,2,4,5]=>5
[6,1,3,2,5,4]=>6
[6,1,3,4,2,5]=>4
[6,1,3,4,5,2]=>5
[6,1,3,5,2,4]=>3
[6,1,3,5,4,2]=>6
[6,1,4,2,3,5]=>4
[6,1,4,2,5,3]=>7
[6,1,4,3,2,5]=>5
[6,1,4,3,5,2]=>6
[6,1,4,5,2,3]=>4
[6,1,4,5,3,2]=>5
[6,1,5,2,3,4]=>5
[6,1,5,2,4,3]=>6
[6,1,5,3,2,4]=>6
[6,1,5,3,4,2]=>5
[6,1,5,4,2,3]=>5
[6,1,5,4,3,2]=>6
[6,2,1,3,4,5]=>5
[6,2,1,3,5,4]=>6
[6,2,1,4,3,5]=>6
[6,2,1,4,5,3]=>5
[6,2,1,5,3,4]=>5
[6,2,1,5,4,3]=>6
[6,2,3,1,4,5]=>4
[6,2,3,1,5,4]=>5
[6,2,3,4,1,5]=>5
[6,2,3,4,5,1]=>4
[6,2,3,5,1,4]=>4
[6,2,3,5,4,1]=>5
[6,2,4,1,3,5]=>3
[6,2,4,1,5,3]=>6
[6,2,4,3,1,5]=>6
[6,2,4,3,5,1]=>5
[6,2,4,5,1,3]=>5
[6,2,4,5,3,1]=>4
[6,2,5,1,3,4]=>4
[6,2,5,1,4,3]=>5
[6,2,5,3,1,4]=>7
[6,2,5,3,4,1]=>4
[6,2,5,4,1,3]=>6
[6,2,5,4,3,1]=>5
[6,3,1,2,4,5]=>4
[6,3,1,2,5,4]=>5
[6,3,1,4,2,5]=>7
[6,3,1,4,5,2]=>4
[6,3,1,5,2,4]=>6
[6,3,1,5,4,2]=>5
[6,3,2,1,4,5]=>5
[6,3,2,1,5,4]=>6
[6,3,2,4,1,5]=>6
[6,3,2,4,5,1]=>5
[6,3,2,5,1,4]=>5
[6,3,2,5,4,1]=>6
[6,3,4,1,2,5]=>4
[6,3,4,1,5,2]=>5
[6,3,4,2,1,5]=>5
[6,3,4,2,5,1]=>4
[6,3,4,5,1,2]=>4
[6,3,4,5,2,1]=>5
[6,3,5,1,2,4]=>5
[6,3,5,1,4,2]=>4
[6,3,5,2,1,4]=>6
[6,3,5,2,4,1]=>3
[6,3,5,4,1,2]=>5
[6,3,5,4,2,1]=>6
[6,4,1,2,3,5]=>5
[6,4,1,2,5,3]=>4
[6,4,1,3,2,5]=>6
[6,4,1,3,5,2]=>3
[6,4,1,5,2,3]=>5
[6,4,1,5,3,2]=>6
[6,4,2,1,3,5]=>6
[6,4,2,1,5,3]=>5
[6,4,2,3,1,5]=>5
[6,4,2,3,5,1]=>4
[6,4,2,5,1,3]=>4
[6,4,2,5,3,1]=>7
[6,4,3,1,2,5]=>5
[6,4,3,1,5,2]=>6
[6,4,3,2,1,5]=>6
[6,4,3,2,5,1]=>5
[6,4,3,5,1,2]=>5
[6,4,3,5,2,1]=>6
[6,4,5,1,2,3]=>4
[6,4,5,1,3,2]=>5
[6,4,5,2,1,3]=>5
[6,4,5,2,3,1]=>4
[6,4,5,3,1,2]=>4
[6,4,5,3,2,1]=>5
[6,5,1,2,3,4]=>4
[6,5,1,2,4,3]=>5
[6,5,1,3,2,4]=>5
[6,5,1,3,4,2]=>4
[6,5,1,4,2,3]=>4
[6,5,1,4,3,2]=>5
[6,5,2,1,3,4]=>5
[6,5,2,1,4,3]=>6
[6,5,2,3,1,4]=>4
[6,5,2,3,4,1]=>5
[6,5,2,4,1,3]=>3
[6,5,2,4,3,1]=>6
[6,5,3,1,2,4]=>4
[6,5,3,1,4,2]=>7
[6,5,3,2,1,4]=>5
[6,5,3,2,4,1]=>6
[6,5,3,4,1,2]=>4
[6,5,3,4,2,1]=>5
[6,5,4,1,2,3]=>5
[6,5,4,1,3,2]=>6
[6,5,4,2,1,3]=>6
[6,5,4,2,3,1]=>5
[6,5,4,3,1,2]=>5
[6,5,4,3,2,1]=>6
[1,2,3,4,5,6,7]=>3
[1,2,3,4,5,7,6]=>4
[1,2,3,4,6,5,7]=>4
[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]=>4
[1,2,3,5,4,7,6]=>5
[1,2,3,5,6,4,7]=>3
[1,2,3,5,6,7,4]=>4
[1,2,3,5,7,4,6]=>2
[1,2,3,5,7,6,4]=>5
[1,2,3,6,4,5,7]=>3
[1,2,3,6,4,7,5]=>6
[1,2,3,6,5,4,7]=>4
[1,2,3,6,5,7,4]=>5
[1,2,3,6,7,4,5]=>3
[1,2,3,6,7,5,4]=>4
[1,2,3,7,4,5,6]=>4
[1,2,3,7,4,6,5]=>5
[1,2,3,7,5,4,6]=>5
[1,2,3,7,5,6,4]=>4
[1,2,3,7,6,4,5]=>4
[1,2,3,7,6,5,4]=>5
[1,2,4,3,5,6,7]=>4
[1,2,4,3,5,7,6]=>5
[1,2,4,3,6,5,7]=>5
[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]=>4
[1,2,4,5,6,7,3]=>3
[1,2,4,5,7,3,6]=>3
[1,2,4,5,7,6,3]=>4
[1,2,4,6,3,5,7]=>2
[1,2,4,6,3,7,5]=>5
[1,2,4,6,5,3,7]=>5
[1,2,4,6,5,7,3]=>4
[1,2,4,6,7,3,5]=>4
[1,2,4,6,7,5,3]=>3
[1,2,4,7,3,5,6]=>3
[1,2,4,7,3,6,5]=>4
[1,2,4,7,5,3,6]=>6
[1,2,4,7,5,6,3]=>3
[1,2,4,7,6,3,5]=>5
[1,2,4,7,6,5,3]=>4
[1,2,5,3,4,6,7]=>3
[1,2,5,3,4,7,6]=>4
[1,2,5,3,6,4,7]=>6
[1,2,5,3,6,7,4]=>3
[1,2,5,3,7,4,6]=>5
[1,2,5,3,7,6,4]=>4
[1,2,5,4,3,6,7]=>4
[1,2,5,4,3,7,6]=>5
[1,2,5,4,6,3,7]=>5
[1,2,5,4,6,7,3]=>4
[1,2,5,4,7,3,6]=>4
[1,2,5,4,7,6,3]=>5
[1,2,5,6,3,4,7]=>3
[1,2,5,6,3,7,4]=>4
[1,2,5,6,4,3,7]=>4
[1,2,5,6,4,7,3]=>3
[1,2,5,6,7,3,4]=>3
[1,2,5,6,7,4,3]=>4
[1,2,5,7,3,4,6]=>4
[1,2,5,7,3,6,4]=>3
[1,2,5,7,4,3,6]=>5
[1,2,5,7,4,6,3]=>2
[1,2,5,7,6,3,4]=>4
[1,2,5,7,6,4,3]=>5
[1,2,6,3,4,5,7]=>4
[1,2,6,3,4,7,5]=>3
[1,2,6,3,5,4,7]=>5
[1,2,6,3,5,7,4]=>2
[1,2,6,3,7,4,5]=>4
[1,2,6,3,7,5,4]=>5
[1,2,6,4,3,5,7]=>5
[1,2,6,4,3,7,5]=>4
[1,2,6,4,5,3,7]=>4
[1,2,6,4,5,7,3]=>3
[1,2,6,4,7,3,5]=>3
[1,2,6,4,7,5,3]=>6
[1,2,6,5,3,4,7]=>4
[1,2,6,5,3,7,4]=>5
[1,2,6,5,4,3,7]=>5
[1,2,6,5,4,7,3]=>4
[1,2,6,5,7,3,4]=>4
[1,2,6,5,7,4,3]=>5
[1,2,6,7,3,4,5]=>3
[1,2,6,7,3,5,4]=>4
[1,2,6,7,4,3,5]=>4
[1,2,6,7,4,5,3]=>3
[1,2,6,7,5,3,4]=>3
[1,2,6,7,5,4,3]=>4
[1,2,7,3,4,5,6]=>3
[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]=>3
[1,2,7,3,6,5,4]=>4
[1,2,7,4,3,5,6]=>4
[1,2,7,4,3,6,5]=>5
[1,2,7,4,5,3,6]=>3
[1,2,7,4,5,6,3]=>4
[1,2,7,4,6,3,5]=>2
[1,2,7,4,6,5,3]=>5
[1,2,7,5,3,4,6]=>3
[1,2,7,5,3,6,4]=>6
[1,2,7,5,4,3,6]=>4
[1,2,7,5,4,6,3]=>5
[1,2,7,5,6,3,4]=>3
[1,2,7,5,6,4,3]=>4
[1,2,7,6,3,4,5]=>4
[1,2,7,6,3,5,4]=>5
[1,2,7,6,4,3,5]=>5
[1,2,7,6,4,5,3]=>4
[1,2,7,6,5,3,4]=>4
[1,2,7,6,5,4,3]=>5
[1,3,2,4,5,6,7]=>4
[1,3,2,4,5,7,6]=>5
[1,3,2,4,6,5,7]=>5
[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]=>5
[1,3,2,5,4,7,6]=>6
[1,3,2,5,6,4,7]=>4
[1,3,2,5,6,7,4]=>5
[1,3,2,5,7,4,6]=>3
[1,3,2,5,7,6,4]=>6
[1,3,2,6,4,5,7]=>4
[1,3,2,6,4,7,5]=>7
[1,3,2,6,5,4,7]=>5
[1,3,2,6,5,7,4]=>6
[1,3,2,6,7,4,5]=>4
[1,3,2,6,7,5,4]=>5
[1,3,2,7,4,5,6]=>5
[1,3,2,7,4,6,5]=>6
[1,3,2,7,5,4,6]=>6
[1,3,2,7,5,6,4]=>5
[1,3,2,7,6,4,5]=>5
[1,3,2,7,6,5,4]=>6
[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]=>3
[1,3,4,2,7,5,6]=>3
[1,3,4,2,7,6,5]=>4
[1,3,4,5,2,6,7]=>4
[1,3,4,5,2,7,6]=>5
[1,3,4,5,6,2,7]=>3
[1,3,4,5,6,7,2]=>4
[1,3,4,5,7,2,6]=>2
[1,3,4,5,7,6,2]=>5
[1,3,4,6,2,5,7]=>3
[1,3,4,6,2,7,5]=>6
[1,3,4,6,5,2,7]=>4
[1,3,4,6,5,7,2]=>5
[1,3,4,6,7,2,5]=>3
[1,3,4,6,7,5,2]=>4
[1,3,4,7,2,5,6]=>4
[1,3,4,7,2,6,5]=>5
[1,3,4,7,5,2,6]=>5
[1,3,4,7,5,6,2]=>4
[1,3,4,7,6,2,5]=>4
[1,3,4,7,6,5,2]=>5
[1,3,5,2,4,6,7]=>2
[1,3,5,2,4,7,6]=>3
[1,3,5,2,6,4,7]=>5
[1,3,5,2,6,7,4]=>2
[1,3,5,2,7,4,6]=>4
[1,3,5,2,7,6,4]=>3
[1,3,5,4,2,6,7]=>5
[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]=>3
[1,3,5,4,7,6,2]=>6
[1,3,5,6,2,4,7]=>4
[1,3,5,6,2,7,4]=>5
[1,3,5,6,4,2,7]=>3
[1,3,5,6,4,7,2]=>4
[1,3,5,6,7,2,4]=>2
[1,3,5,6,7,4,2]=>5
[1,3,5,7,2,4,6]=>5
[1,3,5,7,2,6,4]=>4
[1,3,5,7,4,2,6]=>4
[1,3,5,7,4,6,2]=>3
[1,3,5,7,6,2,4]=>3
[1,3,5,7,6,4,2]=>6
[1,3,6,2,4,5,7]=>3
[1,3,6,2,4,7,5]=>2
[1,3,6,2,5,4,7]=>4
[1,3,6,2,5,7,4]=>1
[1,3,6,2,7,4,5]=>3
[1,3,6,2,7,5,4]=>4
[1,3,6,4,2,5,7]=>6
[1,3,6,4,2,7,5]=>5
[1,3,6,4,5,2,7]=>3
[1,3,6,4,5,7,2]=>4
[1,3,6,4,7,2,5]=>2
[1,3,6,4,7,5,2]=>7
[1,3,6,5,2,4,7]=>5
[1,3,6,5,2,7,4]=>6
[1,3,6,5,4,2,7]=>4
[1,3,6,5,4,7,2]=>5
[1,3,6,5,7,2,4]=>3
[1,3,6,5,7,4,2]=>6
[1,3,6,7,2,4,5]=>4
[1,3,6,7,2,5,4]=>5
[1,3,6,7,4,2,5]=>3
[1,3,6,7,4,5,2]=>4
[1,3,6,7,5,2,4]=>2
[1,3,6,7,5,4,2]=>5
[1,3,7,2,4,5,6]=>2
[1,3,7,2,4,6,5]=>3
[1,3,7,2,5,4,6]=>3
[1,3,7,2,5,6,4]=>2
[1,3,7,2,6,4,5]=>2
[1,3,7,2,6,5,4]=>3
[1,3,7,4,2,5,6]=>5
[1,3,7,4,2,6,5]=>6
[1,3,7,4,5,2,6]=>2
[1,3,7,4,5,6,2]=>5
[1,3,7,4,6,2,5]=>1
[1,3,7,4,6,5,2]=>6
[1,3,7,5,2,4,6]=>4
[1,3,7,5,2,6,4]=>7
[1,3,7,5,4,2,6]=>3
[1,3,7,5,4,6,2]=>6
[1,3,7,5,6,2,4]=>2
[1,3,7,5,6,4,2]=>5
[1,3,7,6,2,4,5]=>5
[1,3,7,6,2,5,4]=>6
[1,3,7,6,4,2,5]=>4
[1,3,7,6,4,5,2]=>5
[1,3,7,6,5,2,4]=>3
[1,3,7,6,5,4,2]=>6
[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]=>3
[1,4,2,3,7,5,6]=>3
[1,4,2,3,7,6,5]=>4
[1,4,2,5,3,6,7]=>6
[1,4,2,5,3,7,6]=>7
[1,4,2,5,6,3,7]=>3
[1,4,2,5,6,7,3]=>6
[1,4,2,5,7,3,6]=>2
[1,4,2,5,7,6,3]=>7
[1,4,2,6,3,5,7]=>5
[1,4,2,6,3,7,5]=>8
[1,4,2,6,5,3,7]=>4
[1,4,2,6,5,7,3]=>7
[1,4,2,6,7,3,5]=>3
[1,4,2,6,7,5,3]=>6
[1,4,2,7,3,5,6]=>6
[1,4,2,7,3,6,5]=>7
[1,4,2,7,5,3,6]=>5
[1,4,2,7,5,6,3]=>6
[1,4,2,7,6,3,5]=>4
[1,4,2,7,6,5,3]=>7
[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]=>4
[1,4,3,2,7,5,6]=>4
[1,4,3,2,7,6,5]=>5
[1,4,3,5,2,6,7]=>5
[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]=>3
[1,4,3,5,7,6,2]=>6
[1,4,3,6,2,5,7]=>4
[1,4,3,6,2,7,5]=>7
[1,4,3,6,5,2,7]=>5
[1,4,3,6,5,7,2]=>6
[1,4,3,6,7,2,5]=>4
[1,4,3,6,7,5,2]=>5
[1,4,3,7,2,5,6]=>5
[1,4,3,7,2,6,5]=>6
[1,4,3,7,5,2,6]=>6
[1,4,3,7,5,6,2]=>5
[1,4,3,7,6,2,5]=>5
[1,4,3,7,6,5,2]=>6
[1,4,5,2,3,6,7]=>3
[1,4,5,2,3,7,6]=>4
[1,4,5,2,6,3,7]=>4
[1,4,5,2,6,7,3]=>3
[1,4,5,2,7,3,6]=>3
[1,4,5,2,7,6,3]=>4
[1,4,5,3,2,6,7]=>4
[1,4,5,3,2,7,6]=>5
[1,4,5,3,6,2,7]=>3
[1,4,5,3,6,7,2]=>4
[1,4,5,3,7,2,6]=>2
[1,4,5,3,7,6,2]=>5
[1,4,5,6,2,3,7]=>3
[1,4,5,6,2,7,3]=>6
[1,4,5,6,3,2,7]=>4
[1,4,5,6,3,7,2]=>5
[1,4,5,6,7,2,3]=>3
[1,4,5,6,7,3,2]=>4
[1,4,5,7,2,3,6]=>4
[1,4,5,7,2,6,3]=>5
[1,4,5,7,3,2,6]=>5
[1,4,5,7,3,6,2]=>4
[1,4,5,7,6,2,3]=>4
[1,4,5,7,6,3,2]=>5
[1,4,6,2,3,5,7]=>4
[1,4,6,2,3,7,5]=>3
[1,4,6,2,5,3,7]=>3
[1,4,6,2,5,7,3]=>2
[1,4,6,2,7,3,5]=>2
[1,4,6,2,7,5,3]=>5
[1,4,6,3,2,5,7]=>5
[1,4,6,3,2,7,5]=>4
[1,4,6,3,5,2,7]=>2
[1,4,6,3,5,7,2]=>3
[1,4,6,3,7,2,5]=>1
[1,4,6,3,7,5,2]=>6
[1,4,6,5,2,3,7]=>4
[1,4,6,5,2,7,3]=>7
[1,4,6,5,3,2,7]=>5
[1,4,6,5,3,7,2]=>6
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Description
The standardized bi-alternating inversion number of a permutation.
The bi-alternating inversion number $i(\pi)$ is $\sum_{1\leq y < x\leq n} (-1)^{y+x} \mathrm{sgn}(\pi(x)-\pi(y))$, see Section 3.1 of [1].
It takes values in between $-\lfloor\frac n2\rfloor^2$ and $\lfloor\frac n2\rfloor^2$, and $i(\pi)+\lfloor\frac n2\rfloor^2$ is even, so we standardize it to $(i(\pi)+\lfloor\frac n2\rfloor^2)/2$.
For odd $n$ the bi-alternating inversion is invariant under rotation, that is, conjugation with the long cycle.
The bi-alternating inversion number $i(\pi)$ is $\sum_{1\leq y < x\leq n} (-1)^{y+x} \mathrm{sgn}(\pi(x)-\pi(y))$, see Section 3.1 of [1].
It takes values in between $-\lfloor\frac n2\rfloor^2$ and $\lfloor\frac n2\rfloor^2$, and $i(\pi)+\lfloor\frac n2\rfloor^2$ is even, so we standardize it to $(i(\pi)+\lfloor\frac n2\rfloor^2)/2$.
For odd $n$ the bi-alternating inversion is invariant under rotation, that is, conjugation with the long cycle.
References
[1] Even-Zohar, C. The Writhe of Permutations and Random Framed Knots arXiv:1511.09469
Code
def statistic(pi):
"""
sage: all(statistic(Permutation([e+1 if e < len(pi) else 1 for e in [pi[-1]] + pi[:-1]])) == statistic(pi) for r in range(1,8,2) for pi in Permutations(r))
True
"""
return 1/2*((len(pi)//2)^2+sum((-1)^(y+x) if pi(x) > pi(y) else -(-1)^(y+x) for x in range(1,len(pi)+1) for y in range(1,x)))
Created
Dec 30, 2016 at 14:38 by Martin Rubey
Updated
Dec 30, 2016 at 14:38 by Martin Rubey
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