Identifier
-
Mp00062:
Permutations
—Lehmer-code to major-code bijection⟶
Permutations
St000004: Permutations ⟶ ℤ
Values
=>
[1]=>[1]=>0
[1,2]=>[1,2]=>0
[2,1]=>[2,1]=>1
[1,2,3]=>[1,2,3]=>0
[1,3,2]=>[3,1,2]=>1
[2,1,3]=>[2,1,3]=>1
[2,3,1]=>[1,3,2]=>2
[3,1,2]=>[2,3,1]=>2
[3,2,1]=>[3,2,1]=>3
[1,2,3,4]=>[1,2,3,4]=>0
[1,2,4,3]=>[4,1,2,3]=>1
[1,3,2,4]=>[3,1,2,4]=>1
[1,3,4,2]=>[2,4,1,3]=>2
[1,4,2,3]=>[3,4,1,2]=>2
[1,4,3,2]=>[4,3,1,2]=>3
[2,1,3,4]=>[2,1,3,4]=>1
[2,1,4,3]=>[1,4,2,3]=>2
[2,3,1,4]=>[1,3,2,4]=>2
[2,3,4,1]=>[1,2,4,3]=>3
[2,4,1,3]=>[1,3,4,2]=>3
[2,4,3,1]=>[4,1,3,2]=>4
[3,1,2,4]=>[2,3,1,4]=>2
[3,1,4,2]=>[4,2,1,3]=>3
[3,2,1,4]=>[3,2,1,4]=>3
[3,2,4,1]=>[2,1,4,3]=>4
[3,4,1,2]=>[3,1,4,2]=>4
[3,4,2,1]=>[1,4,3,2]=>5
[4,1,2,3]=>[2,3,4,1]=>3
[4,1,3,2]=>[4,2,3,1]=>4
[4,2,1,3]=>[3,2,4,1]=>4
[4,2,3,1]=>[2,4,3,1]=>5
[4,3,1,2]=>[3,4,2,1]=>5
[4,3,2,1]=>[4,3,2,1]=>6
[1,2,3,4,5]=>[1,2,3,4,5]=>0
[1,2,3,5,4]=>[5,1,2,3,4]=>1
[1,2,4,3,5]=>[4,1,2,3,5]=>1
[1,2,4,5,3]=>[3,5,1,2,4]=>2
[1,2,5,3,4]=>[4,5,1,2,3]=>2
[1,2,5,4,3]=>[5,4,1,2,3]=>3
[1,3,2,4,5]=>[3,1,2,4,5]=>1
[1,3,2,5,4]=>[2,5,1,3,4]=>2
[1,3,4,2,5]=>[2,4,1,3,5]=>2
[1,3,4,5,2]=>[2,3,5,1,4]=>3
[1,3,5,2,4]=>[2,4,5,1,3]=>3
[1,3,5,4,2]=>[5,2,4,1,3]=>4
[1,4,2,3,5]=>[3,4,1,2,5]=>2
[1,4,2,5,3]=>[5,3,1,2,4]=>3
[1,4,3,2,5]=>[4,3,1,2,5]=>3
[1,4,3,5,2]=>[3,2,5,1,4]=>4
[1,4,5,2,3]=>[4,2,5,1,3]=>4
[1,4,5,3,2]=>[2,5,4,1,3]=>5
[1,5,2,3,4]=>[3,4,5,1,2]=>3
[1,5,2,4,3]=>[5,3,4,1,2]=>4
[1,5,3,2,4]=>[4,3,5,1,2]=>4
[1,5,3,4,2]=>[3,5,4,1,2]=>5
[1,5,4,2,3]=>[4,5,3,1,2]=>5
[1,5,4,3,2]=>[5,4,3,1,2]=>6
[2,1,3,4,5]=>[2,1,3,4,5]=>1
[2,1,3,5,4]=>[1,5,2,3,4]=>2
[2,1,4,3,5]=>[1,4,2,3,5]=>2
[2,1,4,5,3]=>[1,3,5,2,4]=>3
[2,1,5,3,4]=>[1,4,5,2,3]=>3
[2,1,5,4,3]=>[5,1,4,2,3]=>4
[2,3,1,4,5]=>[1,3,2,4,5]=>2
[2,3,1,5,4]=>[1,2,5,3,4]=>3
[2,3,4,1,5]=>[1,2,4,3,5]=>3
[2,3,4,5,1]=>[1,2,3,5,4]=>4
[2,3,5,1,4]=>[1,2,4,5,3]=>4
[2,3,5,4,1]=>[5,1,2,4,3]=>5
[2,4,1,3,5]=>[1,3,4,2,5]=>3
[2,4,1,5,3]=>[5,1,3,2,4]=>4
[2,4,3,1,5]=>[4,1,3,2,5]=>4
[2,4,3,5,1]=>[3,1,2,5,4]=>5
[2,4,5,1,3]=>[4,1,2,5,3]=>5
[2,4,5,3,1]=>[2,5,1,4,3]=>6
[2,5,1,3,4]=>[1,3,4,5,2]=>4
[2,5,1,4,3]=>[5,1,3,4,2]=>5
[2,5,3,1,4]=>[4,1,3,5,2]=>5
[2,5,3,4,1]=>[3,5,1,4,2]=>6
[2,5,4,1,3]=>[4,5,1,3,2]=>6
[2,5,4,3,1]=>[5,4,1,3,2]=>7
[3,1,2,4,5]=>[2,3,1,4,5]=>2
[3,1,2,5,4]=>[5,2,1,3,4]=>3
[3,1,4,2,5]=>[4,2,1,3,5]=>3
[3,1,4,5,2]=>[3,1,5,2,4]=>4
[3,1,5,2,4]=>[4,1,5,2,3]=>4
[3,1,5,4,2]=>[1,5,4,2,3]=>5
[3,2,1,4,5]=>[3,2,1,4,5]=>3
[3,2,1,5,4]=>[2,1,5,3,4]=>4
[3,2,4,1,5]=>[2,1,4,3,5]=>4
[3,2,4,5,1]=>[2,1,3,5,4]=>5
[3,2,5,1,4]=>[2,1,4,5,3]=>5
[3,2,5,4,1]=>[1,5,2,4,3]=>6
[3,4,1,2,5]=>[3,1,4,2,5]=>4
[3,4,1,5,2]=>[1,5,3,2,4]=>5
[3,4,2,1,5]=>[1,4,3,2,5]=>5
[3,4,2,5,1]=>[1,3,2,5,4]=>6
[3,4,5,1,2]=>[1,4,2,5,3]=>6
[3,4,5,2,1]=>[1,2,5,4,3]=>7
[3,5,1,2,4]=>[3,1,4,5,2]=>5
[3,5,1,4,2]=>[1,5,3,4,2]=>6
[3,5,2,1,4]=>[1,4,3,5,2]=>6
[3,5,2,4,1]=>[1,3,5,4,2]=>7
[3,5,4,1,2]=>[1,4,5,3,2]=>7
[3,5,4,2,1]=>[5,1,4,3,2]=>8
[4,1,2,3,5]=>[2,3,4,1,5]=>3
[4,1,2,5,3]=>[5,2,3,1,4]=>4
[4,1,3,2,5]=>[4,2,3,1,5]=>4
[4,1,3,5,2]=>[3,5,2,1,4]=>5
[4,1,5,2,3]=>[4,5,2,1,3]=>5
[4,1,5,3,2]=>[5,4,2,1,3]=>6
[4,2,1,3,5]=>[3,2,4,1,5]=>4
[4,2,1,5,3]=>[2,5,3,1,4]=>5
[4,2,3,1,5]=>[2,4,3,1,5]=>5
[4,2,3,5,1]=>[2,3,1,5,4]=>6
[4,2,5,1,3]=>[2,4,1,5,3]=>6
[4,2,5,3,1]=>[5,2,1,4,3]=>7
[4,3,1,2,5]=>[3,4,2,1,5]=>5
[4,3,1,5,2]=>[5,3,2,1,4]=>6
[4,3,2,1,5]=>[4,3,2,1,5]=>6
[4,3,2,5,1]=>[3,2,1,5,4]=>7
[4,3,5,1,2]=>[4,2,1,5,3]=>7
[4,3,5,2,1]=>[2,1,5,4,3]=>8
[4,5,1,2,3]=>[3,4,1,5,2]=>6
[4,5,1,3,2]=>[5,3,1,4,2]=>7
[4,5,2,1,3]=>[4,3,1,5,2]=>7
[4,5,2,3,1]=>[3,1,5,4,2]=>8
[4,5,3,1,2]=>[4,1,5,3,2]=>8
[4,5,3,2,1]=>[1,5,4,3,2]=>9
[5,1,2,3,4]=>[2,3,4,5,1]=>4
[5,1,2,4,3]=>[5,2,3,4,1]=>5
[5,1,3,2,4]=>[4,2,3,5,1]=>5
[5,1,3,4,2]=>[3,5,2,4,1]=>6
[5,1,4,2,3]=>[4,5,2,3,1]=>6
[5,1,4,3,2]=>[5,4,2,3,1]=>7
[5,2,1,3,4]=>[3,2,4,5,1]=>5
[5,2,1,4,3]=>[2,5,3,4,1]=>6
[5,2,3,1,4]=>[2,4,3,5,1]=>6
[5,2,3,4,1]=>[2,3,5,4,1]=>7
[5,2,4,1,3]=>[2,4,5,3,1]=>7
[5,2,4,3,1]=>[5,2,4,3,1]=>8
[5,3,1,2,4]=>[3,4,2,5,1]=>6
[5,3,1,4,2]=>[5,3,2,4,1]=>7
[5,3,2,1,4]=>[4,3,2,5,1]=>7
[5,3,2,4,1]=>[3,2,5,4,1]=>8
[5,3,4,1,2]=>[4,2,5,3,1]=>8
[5,3,4,2,1]=>[2,5,4,3,1]=>9
[5,4,1,2,3]=>[3,4,5,2,1]=>7
[5,4,1,3,2]=>[5,3,4,2,1]=>8
[5,4,2,1,3]=>[4,3,5,2,1]=>8
[5,4,2,3,1]=>[3,5,4,2,1]=>9
[5,4,3,1,2]=>[4,5,3,2,1]=>9
[5,4,3,2,1]=>[5,4,3,2,1]=>10
[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>[6,1,2,3,4,5]=>1
[1,2,3,5,4,6]=>[5,1,2,3,4,6]=>1
[1,2,3,5,6,4]=>[4,6,1,2,3,5]=>2
[1,2,3,6,4,5]=>[5,6,1,2,3,4]=>2
[1,2,3,6,5,4]=>[6,5,1,2,3,4]=>3
[1,2,4,3,5,6]=>[4,1,2,3,5,6]=>1
[1,2,4,3,6,5]=>[3,6,1,2,4,5]=>2
[1,2,4,5,3,6]=>[3,5,1,2,4,6]=>2
[1,2,4,5,6,3]=>[3,4,6,1,2,5]=>3
[1,2,4,6,3,5]=>[3,5,6,1,2,4]=>3
[1,2,4,6,5,3]=>[6,3,5,1,2,4]=>4
[1,2,5,3,4,6]=>[4,5,1,2,3,6]=>2
[1,2,5,3,6,4]=>[6,4,1,2,3,5]=>3
[1,2,5,4,3,6]=>[5,4,1,2,3,6]=>3
[1,2,5,4,6,3]=>[4,3,6,1,2,5]=>4
[1,2,5,6,3,4]=>[5,3,6,1,2,4]=>4
[1,2,5,6,4,3]=>[3,6,5,1,2,4]=>5
[1,2,6,3,4,5]=>[4,5,6,1,2,3]=>3
[1,2,6,3,5,4]=>[6,4,5,1,2,3]=>4
[1,2,6,4,3,5]=>[5,4,6,1,2,3]=>4
[1,2,6,4,5,3]=>[4,6,5,1,2,3]=>5
[1,2,6,5,3,4]=>[5,6,4,1,2,3]=>5
[1,2,6,5,4,3]=>[6,5,4,1,2,3]=>6
[1,3,2,4,5,6]=>[3,1,2,4,5,6]=>1
[1,3,2,4,6,5]=>[2,6,1,3,4,5]=>2
[1,3,2,5,4,6]=>[2,5,1,3,4,6]=>2
[1,3,2,5,6,4]=>[2,4,6,1,3,5]=>3
[1,3,2,6,4,5]=>[2,5,6,1,3,4]=>3
[1,3,2,6,5,4]=>[6,2,5,1,3,4]=>4
[1,3,4,2,5,6]=>[2,4,1,3,5,6]=>2
[1,3,4,2,6,5]=>[2,3,6,1,4,5]=>3
[1,3,4,5,2,6]=>[2,3,5,1,4,6]=>3
[1,3,4,5,6,2]=>[2,3,4,6,1,5]=>4
[1,3,4,6,2,5]=>[2,3,5,6,1,4]=>4
[1,3,4,6,5,2]=>[6,2,3,5,1,4]=>5
[1,3,5,2,4,6]=>[2,4,5,1,3,6]=>3
[1,3,5,2,6,4]=>[6,2,4,1,3,5]=>4
[1,3,5,4,2,6]=>[5,2,4,1,3,6]=>4
[1,3,5,4,6,2]=>[4,2,3,6,1,5]=>5
[1,3,5,6,2,4]=>[5,2,3,6,1,4]=>5
[1,3,5,6,4,2]=>[3,6,2,5,1,4]=>6
[1,3,6,2,4,5]=>[2,4,5,6,1,3]=>4
[1,3,6,2,5,4]=>[6,2,4,5,1,3]=>5
[1,3,6,4,2,5]=>[5,2,4,6,1,3]=>5
[1,3,6,4,5,2]=>[4,6,2,5,1,3]=>6
[1,3,6,5,2,4]=>[5,6,2,4,1,3]=>6
[1,3,6,5,4,2]=>[6,5,2,4,1,3]=>7
[1,4,2,3,5,6]=>[3,4,1,2,5,6]=>2
[1,4,2,3,6,5]=>[6,3,1,2,4,5]=>3
[1,4,2,5,3,6]=>[5,3,1,2,4,6]=>3
[1,4,2,5,6,3]=>[4,2,6,1,3,5]=>4
[1,4,2,6,3,5]=>[5,2,6,1,3,4]=>4
[1,4,2,6,5,3]=>[2,6,5,1,3,4]=>5
[1,4,3,2,5,6]=>[4,3,1,2,5,6]=>3
[1,4,3,2,6,5]=>[3,2,6,1,4,5]=>4
[1,4,3,5,2,6]=>[3,2,5,1,4,6]=>4
[1,4,3,5,6,2]=>[3,2,4,6,1,5]=>5
[1,4,3,6,2,5]=>[3,2,5,6,1,4]=>5
[1,4,3,6,5,2]=>[2,6,3,5,1,4]=>6
[1,4,5,2,3,6]=>[4,2,5,1,3,6]=>4
[1,4,5,2,6,3]=>[2,6,4,1,3,5]=>5
[1,4,5,3,2,6]=>[2,5,4,1,3,6]=>5
[1,4,5,3,6,2]=>[2,4,3,6,1,5]=>6
[1,4,5,6,2,3]=>[2,5,3,6,1,4]=>6
[1,4,5,6,3,2]=>[2,3,6,5,1,4]=>7
[1,4,6,2,3,5]=>[4,2,5,6,1,3]=>5
[1,4,6,2,5,3]=>[2,6,4,5,1,3]=>6
[1,4,6,3,2,5]=>[2,5,4,6,1,3]=>6
[1,4,6,3,5,2]=>[2,4,6,5,1,3]=>7
[1,4,6,5,2,3]=>[2,5,6,4,1,3]=>7
[1,4,6,5,3,2]=>[6,2,5,4,1,3]=>8
[1,5,2,3,4,6]=>[3,4,5,1,2,6]=>3
[1,5,2,3,6,4]=>[6,3,4,1,2,5]=>4
[1,5,2,4,3,6]=>[5,3,4,1,2,6]=>4
[1,5,2,4,6,3]=>[4,6,3,1,2,5]=>5
[1,5,2,6,3,4]=>[5,6,3,1,2,4]=>5
[1,5,2,6,4,3]=>[6,5,3,1,2,4]=>6
[1,5,3,2,4,6]=>[4,3,5,1,2,6]=>4
[1,5,3,2,6,4]=>[3,6,4,1,2,5]=>5
[1,5,3,4,2,6]=>[3,5,4,1,2,6]=>5
[1,5,3,4,6,2]=>[3,4,2,6,1,5]=>6
[1,5,3,6,2,4]=>[3,5,2,6,1,4]=>6
[1,5,3,6,4,2]=>[6,3,2,5,1,4]=>7
[1,5,4,2,3,6]=>[4,5,3,1,2,6]=>5
[1,5,4,2,6,3]=>[6,4,3,1,2,5]=>6
[1,5,4,3,2,6]=>[5,4,3,1,2,6]=>6
[1,5,4,3,6,2]=>[4,3,2,6,1,5]=>7
[1,5,4,6,2,3]=>[5,3,2,6,1,4]=>7
[1,5,4,6,3,2]=>[3,2,6,5,1,4]=>8
[1,5,6,2,3,4]=>[4,5,2,6,1,3]=>6
[1,5,6,2,4,3]=>[6,4,2,5,1,3]=>7
[1,5,6,3,2,4]=>[5,4,2,6,1,3]=>7
[1,5,6,3,4,2]=>[4,2,6,5,1,3]=>8
[1,5,6,4,2,3]=>[5,2,6,4,1,3]=>8
[1,5,6,4,3,2]=>[2,6,5,4,1,3]=>9
[1,6,2,3,4,5]=>[3,4,5,6,1,2]=>4
[1,6,2,3,5,4]=>[6,3,4,5,1,2]=>5
[1,6,2,4,3,5]=>[5,3,4,6,1,2]=>5
[1,6,2,4,5,3]=>[4,6,3,5,1,2]=>6
[1,6,2,5,3,4]=>[5,6,3,4,1,2]=>6
[1,6,2,5,4,3]=>[6,5,3,4,1,2]=>7
[1,6,3,2,4,5]=>[4,3,5,6,1,2]=>5
[1,6,3,2,5,4]=>[3,6,4,5,1,2]=>6
[1,6,3,4,2,5]=>[3,5,4,6,1,2]=>6
[1,6,3,4,5,2]=>[3,4,6,5,1,2]=>7
[1,6,3,5,2,4]=>[3,5,6,4,1,2]=>7
[1,6,3,5,4,2]=>[6,3,5,4,1,2]=>8
[1,6,4,2,3,5]=>[4,5,3,6,1,2]=>6
[1,6,4,2,5,3]=>[6,4,3,5,1,2]=>7
[1,6,4,3,2,5]=>[5,4,3,6,1,2]=>7
[1,6,4,3,5,2]=>[4,3,6,5,1,2]=>8
[1,6,4,5,2,3]=>[5,3,6,4,1,2]=>8
[1,6,4,5,3,2]=>[3,6,5,4,1,2]=>9
[1,6,5,2,3,4]=>[4,5,6,3,1,2]=>7
[1,6,5,2,4,3]=>[6,4,5,3,1,2]=>8
[1,6,5,3,2,4]=>[5,4,6,3,1,2]=>8
[1,6,5,3,4,2]=>[4,6,5,3,1,2]=>9
[1,6,5,4,2,3]=>[5,6,4,3,1,2]=>9
[1,6,5,4,3,2]=>[6,5,4,3,1,2]=>10
[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>1
[2,1,3,4,6,5]=>[1,6,2,3,4,5]=>2
[2,1,3,5,4,6]=>[1,5,2,3,4,6]=>2
[2,1,3,5,6,4]=>[1,4,6,2,3,5]=>3
[2,1,3,6,4,5]=>[1,5,6,2,3,4]=>3
[2,1,3,6,5,4]=>[6,1,5,2,3,4]=>4
[2,1,4,3,5,6]=>[1,4,2,3,5,6]=>2
[2,1,4,3,6,5]=>[1,3,6,2,4,5]=>3
[2,1,4,5,3,6]=>[1,3,5,2,4,6]=>3
[2,1,4,5,6,3]=>[1,3,4,6,2,5]=>4
[2,1,4,6,3,5]=>[1,3,5,6,2,4]=>4
[2,1,4,6,5,3]=>[6,1,3,5,2,4]=>5
[2,1,5,3,4,6]=>[1,4,5,2,3,6]=>3
[2,1,5,3,6,4]=>[6,1,4,2,3,5]=>4
[2,1,5,4,3,6]=>[5,1,4,2,3,6]=>4
[2,1,5,4,6,3]=>[4,1,3,6,2,5]=>5
[2,1,5,6,3,4]=>[5,1,3,6,2,4]=>5
[2,1,5,6,4,3]=>[3,6,1,5,2,4]=>6
[2,1,6,3,4,5]=>[1,4,5,6,2,3]=>4
[2,1,6,3,5,4]=>[6,1,4,5,2,3]=>5
[2,1,6,4,3,5]=>[5,1,4,6,2,3]=>5
[2,1,6,4,5,3]=>[4,6,1,5,2,3]=>6
[2,1,6,5,3,4]=>[5,6,1,4,2,3]=>6
[2,1,6,5,4,3]=>[6,5,1,4,2,3]=>7
[2,3,1,4,5,6]=>[1,3,2,4,5,6]=>2
[2,3,1,4,6,5]=>[1,2,6,3,4,5]=>3
[2,3,1,5,4,6]=>[1,2,5,3,4,6]=>3
[2,3,1,5,6,4]=>[1,2,4,6,3,5]=>4
[2,3,1,6,4,5]=>[1,2,5,6,3,4]=>4
[2,3,1,6,5,4]=>[6,1,2,5,3,4]=>5
[2,3,4,1,5,6]=>[1,2,4,3,5,6]=>3
[2,3,4,1,6,5]=>[1,2,3,6,4,5]=>4
[2,3,4,5,1,6]=>[1,2,3,5,4,6]=>4
[2,3,4,5,6,1]=>[1,2,3,4,6,5]=>5
[2,3,4,6,1,5]=>[1,2,3,5,6,4]=>5
[2,3,4,6,5,1]=>[6,1,2,3,5,4]=>6
[2,3,5,1,4,6]=>[1,2,4,5,3,6]=>4
[2,3,5,1,6,4]=>[6,1,2,4,3,5]=>5
[2,3,5,4,1,6]=>[5,1,2,4,3,6]=>5
[2,3,5,4,6,1]=>[4,1,2,3,6,5]=>6
[2,3,5,6,1,4]=>[5,1,2,3,6,4]=>6
[2,3,5,6,4,1]=>[3,6,1,2,5,4]=>7
[2,3,6,1,4,5]=>[1,2,4,5,6,3]=>5
[2,3,6,1,5,4]=>[6,1,2,4,5,3]=>6
[2,3,6,4,1,5]=>[5,1,2,4,6,3]=>6
[2,3,6,4,5,1]=>[4,6,1,2,5,3]=>7
[2,3,6,5,1,4]=>[5,6,1,2,4,3]=>7
[2,3,6,5,4,1]=>[6,5,1,2,4,3]=>8
[2,4,1,3,5,6]=>[1,3,4,2,5,6]=>3
[2,4,1,3,6,5]=>[6,1,3,2,4,5]=>4
[2,4,1,5,3,6]=>[5,1,3,2,4,6]=>4
[2,4,1,5,6,3]=>[4,1,2,6,3,5]=>5
[2,4,1,6,3,5]=>[5,1,2,6,3,4]=>5
[2,4,1,6,5,3]=>[2,6,1,5,3,4]=>6
[2,4,3,1,5,6]=>[4,1,3,2,5,6]=>4
[2,4,3,1,6,5]=>[3,1,2,6,4,5]=>5
[2,4,3,5,1,6]=>[3,1,2,5,4,6]=>5
[2,4,3,5,6,1]=>[3,1,2,4,6,5]=>6
[2,4,3,6,1,5]=>[3,1,2,5,6,4]=>6
[2,4,3,6,5,1]=>[2,6,1,3,5,4]=>7
[2,4,5,1,3,6]=>[4,1,2,5,3,6]=>5
[2,4,5,1,6,3]=>[2,6,1,4,3,5]=>6
[2,4,5,3,1,6]=>[2,5,1,4,3,6]=>6
[2,4,5,3,6,1]=>[2,4,1,3,6,5]=>7
[2,4,5,6,1,3]=>[2,5,1,3,6,4]=>7
[2,4,5,6,3,1]=>[2,3,6,1,5,4]=>8
[2,4,6,1,3,5]=>[4,1,2,5,6,3]=>6
[2,4,6,1,5,3]=>[2,6,1,4,5,3]=>7
[2,4,6,3,1,5]=>[2,5,1,4,6,3]=>7
[2,4,6,3,5,1]=>[2,4,6,1,5,3]=>8
[2,4,6,5,1,3]=>[2,5,6,1,4,3]=>8
[2,4,6,5,3,1]=>[6,2,5,1,4,3]=>9
[2,5,1,3,4,6]=>[1,3,4,5,2,6]=>4
[2,5,1,3,6,4]=>[6,1,3,4,2,5]=>5
[2,5,1,4,3,6]=>[5,1,3,4,2,6]=>5
[2,5,1,4,6,3]=>[4,6,1,3,2,5]=>6
[2,5,1,6,3,4]=>[5,6,1,3,2,4]=>6
[2,5,1,6,4,3]=>[6,5,1,3,2,4]=>7
[2,5,3,1,4,6]=>[4,1,3,5,2,6]=>5
[2,5,3,1,6,4]=>[3,6,1,4,2,5]=>6
[2,5,3,4,1,6]=>[3,5,1,4,2,6]=>6
[2,5,3,4,6,1]=>[3,4,1,2,6,5]=>7
[2,5,3,6,1,4]=>[3,5,1,2,6,4]=>7
[2,5,3,6,4,1]=>[6,3,1,2,5,4]=>8
[2,5,4,1,3,6]=>[4,5,1,3,2,6]=>6
[2,5,4,1,6,3]=>[6,4,1,3,2,5]=>7
[2,5,4,3,1,6]=>[5,4,1,3,2,6]=>7
[2,5,4,3,6,1]=>[4,3,1,2,6,5]=>8
[2,5,4,6,1,3]=>[5,3,1,2,6,4]=>8
[2,5,4,6,3,1]=>[3,2,6,1,5,4]=>9
[2,5,6,1,3,4]=>[4,5,1,2,6,3]=>7
[2,5,6,1,4,3]=>[6,4,1,2,5,3]=>8
[2,5,6,3,1,4]=>[5,4,1,2,6,3]=>8
[2,5,6,3,4,1]=>[4,2,6,1,5,3]=>9
[2,5,6,4,1,3]=>[5,2,6,1,4,3]=>9
[2,5,6,4,3,1]=>[2,6,5,1,4,3]=>10
[2,6,1,3,4,5]=>[1,3,4,5,6,2]=>5
[2,6,1,3,5,4]=>[6,1,3,4,5,2]=>6
[2,6,1,4,3,5]=>[5,1,3,4,6,2]=>6
[2,6,1,4,5,3]=>[4,6,1,3,5,2]=>7
[2,6,1,5,3,4]=>[5,6,1,3,4,2]=>7
[2,6,1,5,4,3]=>[6,5,1,3,4,2]=>8
[2,6,3,1,4,5]=>[4,1,3,5,6,2]=>6
[2,6,3,1,5,4]=>[3,6,1,4,5,2]=>7
[2,6,3,4,1,5]=>[3,5,1,4,6,2]=>7
[2,6,3,4,5,1]=>[3,4,6,1,5,2]=>8
[2,6,3,5,1,4]=>[3,5,6,1,4,2]=>8
[2,6,3,5,4,1]=>[6,3,5,1,4,2]=>9
[2,6,4,1,3,5]=>[4,5,1,3,6,2]=>7
[2,6,4,1,5,3]=>[6,4,1,3,5,2]=>8
[2,6,4,3,1,5]=>[5,4,1,3,6,2]=>8
[2,6,4,3,5,1]=>[4,3,6,1,5,2]=>9
[2,6,4,5,1,3]=>[5,3,6,1,4,2]=>9
[2,6,4,5,3,1]=>[3,6,5,1,4,2]=>10
[2,6,5,1,3,4]=>[4,5,6,1,3,2]=>8
[2,6,5,1,4,3]=>[6,4,5,1,3,2]=>9
[2,6,5,3,1,4]=>[5,4,6,1,3,2]=>9
[2,6,5,3,4,1]=>[4,6,5,1,3,2]=>10
[2,6,5,4,1,3]=>[5,6,4,1,3,2]=>10
[2,6,5,4,3,1]=>[6,5,4,1,3,2]=>11
[3,1,2,4,5,6]=>[2,3,1,4,5,6]=>2
[3,1,2,4,6,5]=>[6,2,1,3,4,5]=>3
[3,1,2,5,4,6]=>[5,2,1,3,4,6]=>3
[3,1,2,5,6,4]=>[4,1,6,2,3,5]=>4
[3,1,2,6,4,5]=>[5,1,6,2,3,4]=>4
[3,1,2,6,5,4]=>[1,6,5,2,3,4]=>5
[3,1,4,2,5,6]=>[4,2,1,3,5,6]=>3
[3,1,4,2,6,5]=>[3,1,6,2,4,5]=>4
[3,1,4,5,2,6]=>[3,1,5,2,4,6]=>4
[3,1,4,5,6,2]=>[3,1,4,6,2,5]=>5
[3,1,4,6,2,5]=>[3,1,5,6,2,4]=>5
[3,1,4,6,5,2]=>[1,6,3,5,2,4]=>6
[3,1,5,2,4,6]=>[4,1,5,2,3,6]=>4
[3,1,5,2,6,4]=>[1,6,4,2,3,5]=>5
[3,1,5,4,2,6]=>[1,5,4,2,3,6]=>5
[3,1,5,4,6,2]=>[1,4,3,6,2,5]=>6
[3,1,5,6,2,4]=>[1,5,3,6,2,4]=>6
[3,1,5,6,4,2]=>[1,3,6,5,2,4]=>7
[3,1,6,2,4,5]=>[4,1,5,6,2,3]=>5
[3,1,6,2,5,4]=>[1,6,4,5,2,3]=>6
[3,1,6,4,2,5]=>[1,5,4,6,2,3]=>6
[3,1,6,4,5,2]=>[1,4,6,5,2,3]=>7
[3,1,6,5,2,4]=>[1,5,6,4,2,3]=>7
[3,1,6,5,4,2]=>[6,1,5,4,2,3]=>8
[3,2,1,4,5,6]=>[3,2,1,4,5,6]=>3
[3,2,1,4,6,5]=>[2,1,6,3,4,5]=>4
[3,2,1,5,4,6]=>[2,1,5,3,4,6]=>4
[3,2,1,5,6,4]=>[2,1,4,6,3,5]=>5
[3,2,1,6,4,5]=>[2,1,5,6,3,4]=>5
[3,2,1,6,5,4]=>[1,6,2,5,3,4]=>6
[3,2,4,1,5,6]=>[2,1,4,3,5,6]=>4
[3,2,4,1,6,5]=>[2,1,3,6,4,5]=>5
[3,2,4,5,1,6]=>[2,1,3,5,4,6]=>5
[3,2,4,5,6,1]=>[2,1,3,4,6,5]=>6
[3,2,4,6,1,5]=>[2,1,3,5,6,4]=>6
[3,2,4,6,5,1]=>[1,6,2,3,5,4]=>7
[3,2,5,1,4,6]=>[2,1,4,5,3,6]=>5
[3,2,5,1,6,4]=>[1,6,2,4,3,5]=>6
[3,2,5,4,1,6]=>[1,5,2,4,3,6]=>6
[3,2,5,4,6,1]=>[1,4,2,3,6,5]=>7
[3,2,5,6,1,4]=>[1,5,2,3,6,4]=>7
[3,2,5,6,4,1]=>[1,3,6,2,5,4]=>8
[3,2,6,1,4,5]=>[2,1,4,5,6,3]=>6
[3,2,6,1,5,4]=>[1,6,2,4,5,3]=>7
[3,2,6,4,1,5]=>[1,5,2,4,6,3]=>7
[3,2,6,4,5,1]=>[1,4,6,2,5,3]=>8
[3,2,6,5,1,4]=>[1,5,6,2,4,3]=>8
[3,2,6,5,4,1]=>[6,1,5,2,4,3]=>9
[3,4,1,2,5,6]=>[3,1,4,2,5,6]=>4
[3,4,1,2,6,5]=>[1,6,3,2,4,5]=>5
[3,4,1,5,2,6]=>[1,5,3,2,4,6]=>5
[3,4,1,5,6,2]=>[1,4,2,6,3,5]=>6
[3,4,1,6,2,5]=>[1,5,2,6,3,4]=>6
[3,4,1,6,5,2]=>[1,2,6,5,3,4]=>7
[3,4,2,1,5,6]=>[1,4,3,2,5,6]=>5
[3,4,2,1,6,5]=>[1,3,2,6,4,5]=>6
[3,4,2,5,1,6]=>[1,3,2,5,4,6]=>6
[3,4,2,5,6,1]=>[1,3,2,4,6,5]=>7
[3,4,2,6,1,5]=>[1,3,2,5,6,4]=>7
[3,4,2,6,5,1]=>[1,2,6,3,5,4]=>8
[3,4,5,1,2,6]=>[1,4,2,5,3,6]=>6
[3,4,5,1,6,2]=>[1,2,6,4,3,5]=>7
[3,4,5,2,1,6]=>[1,2,5,4,3,6]=>7
[3,4,5,2,6,1]=>[1,2,4,3,6,5]=>8
[3,4,5,6,1,2]=>[1,2,5,3,6,4]=>8
[3,4,5,6,2,1]=>[1,2,3,6,5,4]=>9
[3,4,6,1,2,5]=>[1,4,2,5,6,3]=>7
[3,4,6,1,5,2]=>[1,2,6,4,5,3]=>8
[3,4,6,2,1,5]=>[1,2,5,4,6,3]=>8
[3,4,6,2,5,1]=>[1,2,4,6,5,3]=>9
[3,4,6,5,1,2]=>[1,2,5,6,4,3]=>9
[3,4,6,5,2,1]=>[6,1,2,5,4,3]=>10
[3,5,1,2,4,6]=>[3,1,4,5,2,6]=>5
[3,5,1,2,6,4]=>[1,6,3,4,2,5]=>6
[3,5,1,4,2,6]=>[1,5,3,4,2,6]=>6
[3,5,1,4,6,2]=>[1,4,6,3,2,5]=>7
[3,5,1,6,2,4]=>[1,5,6,3,2,4]=>7
[3,5,1,6,4,2]=>[6,1,5,3,2,4]=>8
[3,5,2,1,4,6]=>[1,4,3,5,2,6]=>6
[3,5,2,1,6,4]=>[1,3,6,4,2,5]=>7
[3,5,2,4,1,6]=>[1,3,5,4,2,6]=>7
[3,5,2,4,6,1]=>[1,3,4,2,6,5]=>8
[3,5,2,6,1,4]=>[1,3,5,2,6,4]=>8
[3,5,2,6,4,1]=>[6,1,3,2,5,4]=>9
[3,5,4,1,2,6]=>[1,4,5,3,2,6]=>7
[3,5,4,1,6,2]=>[6,1,4,3,2,5]=>8
[3,5,4,2,1,6]=>[5,1,4,3,2,6]=>8
[3,5,4,2,6,1]=>[4,1,3,2,6,5]=>9
[3,5,4,6,1,2]=>[5,1,3,2,6,4]=>9
[3,5,4,6,2,1]=>[3,1,2,6,5,4]=>10
[3,5,6,1,2,4]=>[1,4,5,2,6,3]=>8
[3,5,6,1,4,2]=>[6,1,4,2,5,3]=>9
[3,5,6,2,1,4]=>[5,1,4,2,6,3]=>9
[3,5,6,2,4,1]=>[4,1,2,6,5,3]=>10
[3,5,6,4,1,2]=>[5,1,2,6,4,3]=>10
[3,5,6,4,2,1]=>[2,6,1,5,4,3]=>11
[3,6,1,2,4,5]=>[3,1,4,5,6,2]=>6
[3,6,1,2,5,4]=>[1,6,3,4,5,2]=>7
[3,6,1,4,2,5]=>[1,5,3,4,6,2]=>7
[3,6,1,4,5,2]=>[1,4,6,3,5,2]=>8
[3,6,1,5,2,4]=>[1,5,6,3,4,2]=>8
[3,6,1,5,4,2]=>[6,1,5,3,4,2]=>9
[3,6,2,1,4,5]=>[1,4,3,5,6,2]=>7
[3,6,2,1,5,4]=>[1,3,6,4,5,2]=>8
[3,6,2,4,1,5]=>[1,3,5,4,6,2]=>8
[3,6,2,4,5,1]=>[1,3,4,6,5,2]=>9
[3,6,2,5,1,4]=>[1,3,5,6,4,2]=>9
[3,6,2,5,4,1]=>[6,1,3,5,4,2]=>10
[3,6,4,1,2,5]=>[1,4,5,3,6,2]=>8
[3,6,4,1,5,2]=>[6,1,4,3,5,2]=>9
[3,6,4,2,1,5]=>[5,1,4,3,6,2]=>9
[3,6,4,2,5,1]=>[4,1,3,6,5,2]=>10
[3,6,4,5,1,2]=>[5,1,3,6,4,2]=>10
[3,6,4,5,2,1]=>[3,6,1,5,4,2]=>11
[3,6,5,1,2,4]=>[1,4,5,6,3,2]=>9
[3,6,5,1,4,2]=>[6,1,4,5,3,2]=>10
[3,6,5,2,1,4]=>[5,1,4,6,3,2]=>10
[3,6,5,2,4,1]=>[4,6,1,5,3,2]=>11
[3,6,5,4,1,2]=>[5,6,1,4,3,2]=>11
[3,6,5,4,2,1]=>[6,5,1,4,3,2]=>12
[4,1,2,3,5,6]=>[2,3,4,1,5,6]=>3
[4,1,2,3,6,5]=>[6,2,3,1,4,5]=>4
[4,1,2,5,3,6]=>[5,2,3,1,4,6]=>4
[4,1,2,5,6,3]=>[4,6,2,1,3,5]=>5
[4,1,2,6,3,5]=>[5,6,2,1,3,4]=>5
[4,1,2,6,5,3]=>[6,5,2,1,3,4]=>6
[4,1,3,2,5,6]=>[4,2,3,1,5,6]=>4
[4,1,3,2,6,5]=>[3,6,2,1,4,5]=>5
[4,1,3,5,2,6]=>[3,5,2,1,4,6]=>5
[4,1,3,5,6,2]=>[3,4,1,6,2,5]=>6
[4,1,3,6,2,5]=>[3,5,1,6,2,4]=>6
[4,1,3,6,5,2]=>[6,3,1,5,2,4]=>7
[4,1,5,2,3,6]=>[4,5,2,1,3,6]=>5
[4,1,5,2,6,3]=>[6,4,2,1,3,5]=>6
[4,1,5,3,2,6]=>[5,4,2,1,3,6]=>6
[4,1,5,3,6,2]=>[4,3,1,6,2,5]=>7
[4,1,5,6,2,3]=>[5,3,1,6,2,4]=>7
[4,1,5,6,3,2]=>[3,1,6,5,2,4]=>8
[4,1,6,2,3,5]=>[4,5,1,6,2,3]=>6
[4,1,6,2,5,3]=>[6,4,1,5,2,3]=>7
[4,1,6,3,2,5]=>[5,4,1,6,2,3]=>7
[4,1,6,3,5,2]=>[4,1,6,5,2,3]=>8
[4,1,6,5,2,3]=>[5,1,6,4,2,3]=>8
[4,1,6,5,3,2]=>[1,6,5,4,2,3]=>9
[4,2,1,3,5,6]=>[3,2,4,1,5,6]=>4
[4,2,1,3,6,5]=>[2,6,3,1,4,5]=>5
[4,2,1,5,3,6]=>[2,5,3,1,4,6]=>5
[4,2,1,5,6,3]=>[2,4,1,6,3,5]=>6
[4,2,1,6,3,5]=>[2,5,1,6,3,4]=>6
[4,2,1,6,5,3]=>[6,2,1,5,3,4]=>7
[4,2,3,1,5,6]=>[2,4,3,1,5,6]=>5
[4,2,3,1,6,5]=>[2,3,1,6,4,5]=>6
[4,2,3,5,1,6]=>[2,3,1,5,4,6]=>6
[4,2,3,5,6,1]=>[2,3,1,4,6,5]=>7
[4,2,3,6,1,5]=>[2,3,1,5,6,4]=>7
[4,2,3,6,5,1]=>[6,2,1,3,5,4]=>8
[4,2,5,1,3,6]=>[2,4,1,5,3,6]=>6
[4,2,5,1,6,3]=>[6,2,1,4,3,5]=>7
[4,2,5,3,1,6]=>[5,2,1,4,3,6]=>7
[4,2,5,3,6,1]=>[4,2,1,3,6,5]=>8
[4,2,5,6,1,3]=>[5,2,1,3,6,4]=>8
[4,2,5,6,3,1]=>[3,1,6,2,5,4]=>9
[4,2,6,1,3,5]=>[2,4,1,5,6,3]=>7
[4,2,6,1,5,3]=>[6,2,1,4,5,3]=>8
[4,2,6,3,1,5]=>[5,2,1,4,6,3]=>8
[4,2,6,3,5,1]=>[4,1,6,2,5,3]=>9
[4,2,6,5,1,3]=>[5,1,6,2,4,3]=>9
[4,2,6,5,3,1]=>[1,6,5,2,4,3]=>10
[4,3,1,2,5,6]=>[3,4,2,1,5,6]=>5
[4,3,1,2,6,5]=>[6,3,2,1,4,5]=>6
[4,3,1,5,2,6]=>[5,3,2,1,4,6]=>6
[4,3,1,5,6,2]=>[4,2,1,6,3,5]=>7
[4,3,1,6,2,5]=>[5,2,1,6,3,4]=>7
[4,3,1,6,5,2]=>[2,1,6,5,3,4]=>8
[4,3,2,1,5,6]=>[4,3,2,1,5,6]=>6
[4,3,2,1,6,5]=>[3,2,1,6,4,5]=>7
[4,3,2,5,1,6]=>[3,2,1,5,4,6]=>7
[4,3,2,5,6,1]=>[3,2,1,4,6,5]=>8
[4,3,2,6,1,5]=>[3,2,1,5,6,4]=>8
[4,3,2,6,5,1]=>[2,1,6,3,5,4]=>9
[4,3,5,1,2,6]=>[4,2,1,5,3,6]=>7
[4,3,5,1,6,2]=>[2,1,6,4,3,5]=>8
[4,3,5,2,1,6]=>[2,1,5,4,3,6]=>8
[4,3,5,2,6,1]=>[2,1,4,3,6,5]=>9
[4,3,5,6,1,2]=>[2,1,5,3,6,4]=>9
[4,3,5,6,2,1]=>[2,1,3,6,5,4]=>10
[4,3,6,1,2,5]=>[4,2,1,5,6,3]=>8
[4,3,6,1,5,2]=>[2,1,6,4,5,3]=>9
[4,3,6,2,1,5]=>[2,1,5,4,6,3]=>9
[4,3,6,2,5,1]=>[2,1,4,6,5,3]=>10
[4,3,6,5,1,2]=>[2,1,5,6,4,3]=>10
[4,3,6,5,2,1]=>[1,6,2,5,4,3]=>11
[4,5,1,2,3,6]=>[3,4,1,5,2,6]=>6
[4,5,1,2,6,3]=>[6,3,1,4,2,5]=>7
[4,5,1,3,2,6]=>[5,3,1,4,2,6]=>7
[4,5,1,3,6,2]=>[4,1,6,3,2,5]=>8
[4,5,1,6,2,3]=>[5,1,6,3,2,4]=>8
[4,5,1,6,3,2]=>[1,6,5,3,2,4]=>9
[4,5,2,1,3,6]=>[4,3,1,5,2,6]=>7
[4,5,2,1,6,3]=>[3,1,6,4,2,5]=>8
[4,5,2,3,1,6]=>[3,1,5,4,2,6]=>8
[4,5,2,3,6,1]=>[3,1,4,2,6,5]=>9
[4,5,2,6,1,3]=>[3,1,5,2,6,4]=>9
[4,5,2,6,3,1]=>[1,6,3,2,5,4]=>10
[4,5,3,1,2,6]=>[4,1,5,3,2,6]=>8
[4,5,3,1,6,2]=>[1,6,4,3,2,5]=>9
[4,5,3,2,1,6]=>[1,5,4,3,2,6]=>9
[4,5,3,2,6,1]=>[1,4,3,2,6,5]=>10
[4,5,3,6,1,2]=>[1,5,3,2,6,4]=>10
[4,5,3,6,2,1]=>[1,3,2,6,5,4]=>11
[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>9
[4,5,6,1,3,2]=>[1,6,4,2,5,3]=>10
[4,5,6,2,1,3]=>[1,5,4,2,6,3]=>10
[4,5,6,2,3,1]=>[1,4,2,6,5,3]=>11
[4,5,6,3,1,2]=>[1,5,2,6,4,3]=>11
[4,5,6,3,2,1]=>[1,2,6,5,4,3]=>12
[4,6,1,2,3,5]=>[3,4,1,5,6,2]=>7
[4,6,1,2,5,3]=>[6,3,1,4,5,2]=>8
[4,6,1,3,2,5]=>[5,3,1,4,6,2]=>8
[4,6,1,3,5,2]=>[4,1,6,3,5,2]=>9
[4,6,1,5,2,3]=>[5,1,6,3,4,2]=>9
[4,6,1,5,3,2]=>[1,6,5,3,4,2]=>10
[4,6,2,1,3,5]=>[4,3,1,5,6,2]=>8
[4,6,2,1,5,3]=>[3,1,6,4,5,2]=>9
[4,6,2,3,1,5]=>[3,1,5,4,6,2]=>9
[4,6,2,3,5,1]=>[3,1,4,6,5,2]=>10
[4,6,2,5,1,3]=>[3,1,5,6,4,2]=>10
[4,6,2,5,3,1]=>[1,6,3,5,4,2]=>11
[4,6,3,1,2,5]=>[4,1,5,3,6,2]=>9
[4,6,3,1,5,2]=>[1,6,4,3,5,2]=>10
[4,6,3,2,1,5]=>[1,5,4,3,6,2]=>10
[4,6,3,2,5,1]=>[1,4,3,6,5,2]=>11
[4,6,3,5,1,2]=>[1,5,3,6,4,2]=>11
[4,6,3,5,2,1]=>[1,3,6,5,4,2]=>12
[4,6,5,1,2,3]=>[4,1,5,6,3,2]=>10
[4,6,5,1,3,2]=>[1,6,4,5,3,2]=>11
[4,6,5,2,1,3]=>[1,5,4,6,3,2]=>11
[4,6,5,2,3,1]=>[1,4,6,5,3,2]=>12
[4,6,5,3,1,2]=>[1,5,6,4,3,2]=>12
[4,6,5,3,2,1]=>[6,1,5,4,3,2]=>13
[5,1,2,3,4,6]=>[2,3,4,5,1,6]=>4
[5,1,2,3,6,4]=>[6,2,3,4,1,5]=>5
[5,1,2,4,3,6]=>[5,2,3,4,1,6]=>5
[5,1,2,4,6,3]=>[4,6,2,3,1,5]=>6
[5,1,2,6,3,4]=>[5,6,2,3,1,4]=>6
[5,1,2,6,4,3]=>[6,5,2,3,1,4]=>7
[5,1,3,2,4,6]=>[4,2,3,5,1,6]=>5
[5,1,3,2,6,4]=>[3,6,2,4,1,5]=>6
[5,1,3,4,2,6]=>[3,5,2,4,1,6]=>6
[5,1,3,4,6,2]=>[3,4,6,2,1,5]=>7
[5,1,3,6,2,4]=>[3,5,6,2,1,4]=>7
[5,1,3,6,4,2]=>[6,3,5,2,1,4]=>8
[5,1,4,2,3,6]=>[4,5,2,3,1,6]=>6
[5,1,4,2,6,3]=>[6,4,2,3,1,5]=>7
[5,1,4,3,2,6]=>[5,4,2,3,1,6]=>7
[5,1,4,3,6,2]=>[4,3,6,2,1,5]=>8
[5,1,4,6,2,3]=>[5,3,6,2,1,4]=>8
[5,1,4,6,3,2]=>[3,6,5,2,1,4]=>9
[5,1,6,2,3,4]=>[4,5,6,2,1,3]=>7
[5,1,6,2,4,3]=>[6,4,5,2,1,3]=>8
[5,1,6,3,2,4]=>[5,4,6,2,1,3]=>8
[5,1,6,3,4,2]=>[4,6,5,2,1,3]=>9
[5,1,6,4,2,3]=>[5,6,4,2,1,3]=>9
[5,1,6,4,3,2]=>[6,5,4,2,1,3]=>10
[5,2,1,3,4,6]=>[3,2,4,5,1,6]=>5
[5,2,1,3,6,4]=>[2,6,3,4,1,5]=>6
[5,2,1,4,3,6]=>[2,5,3,4,1,6]=>6
[5,2,1,4,6,3]=>[2,4,6,3,1,5]=>7
[5,2,1,6,3,4]=>[2,5,6,3,1,4]=>7
[5,2,1,6,4,3]=>[6,2,5,3,1,4]=>8
[5,2,3,1,4,6]=>[2,4,3,5,1,6]=>6
[5,2,3,1,6,4]=>[2,3,6,4,1,5]=>7
[5,2,3,4,1,6]=>[2,3,5,4,1,6]=>7
[5,2,3,4,6,1]=>[2,3,4,1,6,5]=>8
[5,2,3,6,1,4]=>[2,3,5,1,6,4]=>8
[5,2,3,6,4,1]=>[6,2,3,1,5,4]=>9
[5,2,4,1,3,6]=>[2,4,5,3,1,6]=>7
[5,2,4,1,6,3]=>[6,2,4,3,1,5]=>8
[5,2,4,3,1,6]=>[5,2,4,3,1,6]=>8
[5,2,4,3,6,1]=>[4,2,3,1,6,5]=>9
[5,2,4,6,1,3]=>[5,2,3,1,6,4]=>9
[5,2,4,6,3,1]=>[3,6,2,1,5,4]=>10
[5,2,6,1,3,4]=>[2,4,5,1,6,3]=>8
[5,2,6,1,4,3]=>[6,2,4,1,5,3]=>9
[5,2,6,3,1,4]=>[5,2,4,1,6,3]=>9
[5,2,6,3,4,1]=>[4,6,2,1,5,3]=>10
[5,2,6,4,1,3]=>[5,6,2,1,4,3]=>10
[5,2,6,4,3,1]=>[6,5,2,1,4,3]=>11
[5,3,1,2,4,6]=>[3,4,2,5,1,6]=>6
[5,3,1,2,6,4]=>[6,3,2,4,1,5]=>7
[5,3,1,4,2,6]=>[5,3,2,4,1,6]=>7
[5,3,1,4,6,2]=>[4,2,6,3,1,5]=>8
[5,3,1,6,2,4]=>[5,2,6,3,1,4]=>8
[5,3,1,6,4,2]=>[2,6,5,3,1,4]=>9
[5,3,2,1,4,6]=>[4,3,2,5,1,6]=>7
[5,3,2,1,6,4]=>[3,2,6,4,1,5]=>8
[5,3,2,4,1,6]=>[3,2,5,4,1,6]=>8
[5,3,2,4,6,1]=>[3,2,4,1,6,5]=>9
[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>9
[5,3,2,6,4,1]=>[2,6,3,1,5,4]=>10
[5,3,4,1,2,6]=>[4,2,5,3,1,6]=>8
[5,3,4,1,6,2]=>[2,6,4,3,1,5]=>9
[5,3,4,2,1,6]=>[2,5,4,3,1,6]=>9
[5,3,4,2,6,1]=>[2,4,3,1,6,5]=>10
[5,3,4,6,1,2]=>[2,5,3,1,6,4]=>10
[5,3,4,6,2,1]=>[2,3,1,6,5,4]=>11
[5,3,6,1,2,4]=>[4,2,5,1,6,3]=>9
[5,3,6,1,4,2]=>[2,6,4,1,5,3]=>10
[5,3,6,2,1,4]=>[2,5,4,1,6,3]=>10
[5,3,6,2,4,1]=>[2,4,1,6,5,3]=>11
[5,3,6,4,1,2]=>[2,5,1,6,4,3]=>11
[5,3,6,4,2,1]=>[6,2,1,5,4,3]=>12
[5,4,1,2,3,6]=>[3,4,5,2,1,6]=>7
[5,4,1,2,6,3]=>[6,3,4,2,1,5]=>8
[5,4,1,3,2,6]=>[5,3,4,2,1,6]=>8
[5,4,1,3,6,2]=>[4,6,3,2,1,5]=>9
[5,4,1,6,2,3]=>[5,6,3,2,1,4]=>9
[5,4,1,6,3,2]=>[6,5,3,2,1,4]=>10
[5,4,2,1,3,6]=>[4,3,5,2,1,6]=>8
[5,4,2,1,6,3]=>[3,6,4,2,1,5]=>9
[5,4,2,3,1,6]=>[3,5,4,2,1,6]=>9
[5,4,2,3,6,1]=>[3,4,2,1,6,5]=>10
[5,4,2,6,1,3]=>[3,5,2,1,6,4]=>10
[5,4,2,6,3,1]=>[6,3,2,1,5,4]=>11
[5,4,3,1,2,6]=>[4,5,3,2,1,6]=>9
[5,4,3,1,6,2]=>[6,4,3,2,1,5]=>10
[5,4,3,2,1,6]=>[5,4,3,2,1,6]=>10
[5,4,3,2,6,1]=>[4,3,2,1,6,5]=>11
[5,4,3,6,1,2]=>[5,3,2,1,6,4]=>11
[5,4,3,6,2,1]=>[3,2,1,6,5,4]=>12
[5,4,6,1,2,3]=>[4,5,2,1,6,3]=>10
[5,4,6,1,3,2]=>[6,4,2,1,5,3]=>11
[5,4,6,2,1,3]=>[5,4,2,1,6,3]=>11
[5,4,6,2,3,1]=>[4,2,1,6,5,3]=>12
[5,4,6,3,1,2]=>[5,2,1,6,4,3]=>12
[5,4,6,3,2,1]=>[2,1,6,5,4,3]=>13
[5,6,1,2,3,4]=>[3,4,5,1,6,2]=>8
[5,6,1,2,4,3]=>[6,3,4,1,5,2]=>9
[5,6,1,3,2,4]=>[5,3,4,1,6,2]=>9
[5,6,1,3,4,2]=>[4,6,3,1,5,2]=>10
[5,6,1,4,2,3]=>[5,6,3,1,4,2]=>10
[5,6,1,4,3,2]=>[6,5,3,1,4,2]=>11
[5,6,2,1,3,4]=>[4,3,5,1,6,2]=>9
[5,6,2,1,4,3]=>[3,6,4,1,5,2]=>10
[5,6,2,3,1,4]=>[3,5,4,1,6,2]=>10
[5,6,2,3,4,1]=>[3,4,1,6,5,2]=>11
[5,6,2,4,1,3]=>[3,5,1,6,4,2]=>11
[5,6,2,4,3,1]=>[6,3,1,5,4,2]=>12
[5,6,3,1,2,4]=>[4,5,3,1,6,2]=>10
[5,6,3,1,4,2]=>[6,4,3,1,5,2]=>11
[5,6,3,2,1,4]=>[5,4,3,1,6,2]=>11
[5,6,3,2,4,1]=>[4,3,1,6,5,2]=>12
[5,6,3,4,1,2]=>[5,3,1,6,4,2]=>12
[5,6,3,4,2,1]=>[3,1,6,5,4,2]=>13
[5,6,4,1,2,3]=>[4,5,1,6,3,2]=>11
[5,6,4,1,3,2]=>[6,4,1,5,3,2]=>12
[5,6,4,2,1,3]=>[5,4,1,6,3,2]=>12
[5,6,4,2,3,1]=>[4,1,6,5,3,2]=>13
[5,6,4,3,1,2]=>[5,1,6,4,3,2]=>13
[5,6,4,3,2,1]=>[1,6,5,4,3,2]=>14
[6,1,2,3,4,5]=>[2,3,4,5,6,1]=>5
[6,1,2,3,5,4]=>[6,2,3,4,5,1]=>6
[6,1,2,4,3,5]=>[5,2,3,4,6,1]=>6
[6,1,2,4,5,3]=>[4,6,2,3,5,1]=>7
[6,1,2,5,3,4]=>[5,6,2,3,4,1]=>7
[6,1,2,5,4,3]=>[6,5,2,3,4,1]=>8
[6,1,3,2,4,5]=>[4,2,3,5,6,1]=>6
[6,1,3,2,5,4]=>[3,6,2,4,5,1]=>7
[6,1,3,4,2,5]=>[3,5,2,4,6,1]=>7
[6,1,3,4,5,2]=>[3,4,6,2,5,1]=>8
[6,1,3,5,2,4]=>[3,5,6,2,4,1]=>8
[6,1,3,5,4,2]=>[6,3,5,2,4,1]=>9
[6,1,4,2,3,5]=>[4,5,2,3,6,1]=>7
[6,1,4,2,5,3]=>[6,4,2,3,5,1]=>8
[6,1,4,3,2,5]=>[5,4,2,3,6,1]=>8
[6,1,4,3,5,2]=>[4,3,6,2,5,1]=>9
[6,1,4,5,2,3]=>[5,3,6,2,4,1]=>9
[6,1,4,5,3,2]=>[3,6,5,2,4,1]=>10
[6,1,5,2,3,4]=>[4,5,6,2,3,1]=>8
[6,1,5,2,4,3]=>[6,4,5,2,3,1]=>9
[6,1,5,3,2,4]=>[5,4,6,2,3,1]=>9
[6,1,5,3,4,2]=>[4,6,5,2,3,1]=>10
[6,1,5,4,2,3]=>[5,6,4,2,3,1]=>10
[6,1,5,4,3,2]=>[6,5,4,2,3,1]=>11
[6,2,1,3,4,5]=>[3,2,4,5,6,1]=>6
[6,2,1,3,5,4]=>[2,6,3,4,5,1]=>7
[6,2,1,4,3,5]=>[2,5,3,4,6,1]=>7
[6,2,1,4,5,3]=>[2,4,6,3,5,1]=>8
[6,2,1,5,3,4]=>[2,5,6,3,4,1]=>8
[6,2,1,5,4,3]=>[6,2,5,3,4,1]=>9
[6,2,3,1,4,5]=>[2,4,3,5,6,1]=>7
[6,2,3,1,5,4]=>[2,3,6,4,5,1]=>8
[6,2,3,4,1,5]=>[2,3,5,4,6,1]=>8
[6,2,3,4,5,1]=>[2,3,4,6,5,1]=>9
[6,2,3,5,1,4]=>[2,3,5,6,4,1]=>9
[6,2,3,5,4,1]=>[6,2,3,5,4,1]=>10
[6,2,4,1,3,5]=>[2,4,5,3,6,1]=>8
[6,2,4,1,5,3]=>[6,2,4,3,5,1]=>9
[6,2,4,3,1,5]=>[5,2,4,3,6,1]=>9
[6,2,4,3,5,1]=>[4,2,3,6,5,1]=>10
[6,2,4,5,1,3]=>[5,2,3,6,4,1]=>10
[6,2,4,5,3,1]=>[3,6,2,5,4,1]=>11
[6,2,5,1,3,4]=>[2,4,5,6,3,1]=>9
[6,2,5,1,4,3]=>[6,2,4,5,3,1]=>10
[6,2,5,3,1,4]=>[5,2,4,6,3,1]=>10
[6,2,5,3,4,1]=>[4,6,2,5,3,1]=>11
[6,2,5,4,1,3]=>[5,6,2,4,3,1]=>11
[6,2,5,4,3,1]=>[6,5,2,4,3,1]=>12
[6,3,1,2,4,5]=>[3,4,2,5,6,1]=>7
[6,3,1,2,5,4]=>[6,3,2,4,5,1]=>8
[6,3,1,4,2,5]=>[5,3,2,4,6,1]=>8
[6,3,1,4,5,2]=>[4,2,6,3,5,1]=>9
[6,3,1,5,2,4]=>[5,2,6,3,4,1]=>9
[6,3,1,5,4,2]=>[2,6,5,3,4,1]=>10
[6,3,2,1,4,5]=>[4,3,2,5,6,1]=>8
[6,3,2,1,5,4]=>[3,2,6,4,5,1]=>9
[6,3,2,4,1,5]=>[3,2,5,4,6,1]=>9
[6,3,2,4,5,1]=>[3,2,4,6,5,1]=>10
[6,3,2,5,1,4]=>[3,2,5,6,4,1]=>10
[6,3,2,5,4,1]=>[2,6,3,5,4,1]=>11
[6,3,4,1,2,5]=>[4,2,5,3,6,1]=>9
[6,3,4,1,5,2]=>[2,6,4,3,5,1]=>10
[6,3,4,2,1,5]=>[2,5,4,3,6,1]=>10
[6,3,4,2,5,1]=>[2,4,3,6,5,1]=>11
[6,3,4,5,1,2]=>[2,5,3,6,4,1]=>11
[6,3,4,5,2,1]=>[2,3,6,5,4,1]=>12
[6,3,5,1,2,4]=>[4,2,5,6,3,1]=>10
[6,3,5,1,4,2]=>[2,6,4,5,3,1]=>11
[6,3,5,2,1,4]=>[2,5,4,6,3,1]=>11
[6,3,5,2,4,1]=>[2,4,6,5,3,1]=>12
[6,3,5,4,1,2]=>[2,5,6,4,3,1]=>12
[6,3,5,4,2,1]=>[6,2,5,4,3,1]=>13
[6,4,1,2,3,5]=>[3,4,5,2,6,1]=>8
[6,4,1,2,5,3]=>[6,3,4,2,5,1]=>9
[6,4,1,3,2,5]=>[5,3,4,2,6,1]=>9
[6,4,1,3,5,2]=>[4,6,3,2,5,1]=>10
[6,4,1,5,2,3]=>[5,6,3,2,4,1]=>10
[6,4,1,5,3,2]=>[6,5,3,2,4,1]=>11
[6,4,2,1,3,5]=>[4,3,5,2,6,1]=>9
[6,4,2,1,5,3]=>[3,6,4,2,5,1]=>10
[6,4,2,3,1,5]=>[3,5,4,2,6,1]=>10
[6,4,2,3,5,1]=>[3,4,2,6,5,1]=>11
[6,4,2,5,1,3]=>[3,5,2,6,4,1]=>11
[6,4,2,5,3,1]=>[6,3,2,5,4,1]=>12
[6,4,3,1,2,5]=>[4,5,3,2,6,1]=>10
[6,4,3,1,5,2]=>[6,4,3,2,5,1]=>11
[6,4,3,2,1,5]=>[5,4,3,2,6,1]=>11
[6,4,3,2,5,1]=>[4,3,2,6,5,1]=>12
[6,4,3,5,1,2]=>[5,3,2,6,4,1]=>12
[6,4,3,5,2,1]=>[3,2,6,5,4,1]=>13
[6,4,5,1,2,3]=>[4,5,2,6,3,1]=>11
[6,4,5,1,3,2]=>[6,4,2,5,3,1]=>12
[6,4,5,2,1,3]=>[5,4,2,6,3,1]=>12
[6,4,5,2,3,1]=>[4,2,6,5,3,1]=>13
[6,4,5,3,1,2]=>[5,2,6,4,3,1]=>13
[6,4,5,3,2,1]=>[2,6,5,4,3,1]=>14
[6,5,1,2,3,4]=>[3,4,5,6,2,1]=>9
[6,5,1,2,4,3]=>[6,3,4,5,2,1]=>10
[6,5,1,3,2,4]=>[5,3,4,6,2,1]=>10
[6,5,1,3,4,2]=>[4,6,3,5,2,1]=>11
[6,5,1,4,2,3]=>[5,6,3,4,2,1]=>11
[6,5,1,4,3,2]=>[6,5,3,4,2,1]=>12
[6,5,2,1,3,4]=>[4,3,5,6,2,1]=>10
[6,5,2,1,4,3]=>[3,6,4,5,2,1]=>11
[6,5,2,3,1,4]=>[3,5,4,6,2,1]=>11
[6,5,2,3,4,1]=>[3,4,6,5,2,1]=>12
[6,5,2,4,1,3]=>[3,5,6,4,2,1]=>12
[6,5,2,4,3,1]=>[6,3,5,4,2,1]=>13
[6,5,3,1,2,4]=>[4,5,3,6,2,1]=>11
[6,5,3,1,4,2]=>[6,4,3,5,2,1]=>12
[6,5,3,2,1,4]=>[5,4,3,6,2,1]=>12
[6,5,3,2,4,1]=>[4,3,6,5,2,1]=>13
[6,5,3,4,1,2]=>[5,3,6,4,2,1]=>13
[6,5,3,4,2,1]=>[3,6,5,4,2,1]=>14
[6,5,4,1,2,3]=>[4,5,6,3,2,1]=>12
[6,5,4,1,3,2]=>[6,4,5,3,2,1]=>13
[6,5,4,2,1,3]=>[5,4,6,3,2,1]=>13
[6,5,4,2,3,1]=>[4,6,5,3,2,1]=>14
[6,5,4,3,1,2]=>[5,6,4,3,2,1]=>14
[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>15
[]=>[]=>0
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Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called Mahonian statistic.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called Mahonian statistic.
Map
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
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