Identifier
- St000029: Permutations ⟶ ℤ
Values
=>
[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]=>2
[1,2,3,4]=>0
[1,2,4,3]=>1
[1,3,2,4]=>1
[1,3,4,2]=>2
[1,4,2,3]=>2
[1,4,3,2]=>2
[2,1,3,4]=>1
[2,1,4,3]=>2
[2,3,1,4]=>2
[2,3,4,1]=>3
[2,4,1,3]=>3
[2,4,3,1]=>3
[3,1,2,4]=>2
[3,1,4,2]=>3
[3,2,1,4]=>2
[3,2,4,1]=>3
[3,4,1,2]=>4
[3,4,2,1]=>4
[4,1,2,3]=>3
[4,1,3,2]=>3
[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]=>1
[1,2,4,3,5]=>1
[1,2,4,5,3]=>2
[1,2,5,3,4]=>2
[1,2,5,4,3]=>2
[1,3,2,4,5]=>1
[1,3,2,5,4]=>2
[1,3,4,2,5]=>2
[1,3,4,5,2]=>3
[1,3,5,2,4]=>3
[1,3,5,4,2]=>3
[1,4,2,3,5]=>2
[1,4,2,5,3]=>3
[1,4,3,2,5]=>2
[1,4,3,5,2]=>3
[1,4,5,2,3]=>4
[1,4,5,3,2]=>4
[1,5,2,3,4]=>3
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>3
[1,5,4,2,3]=>4
[1,5,4,3,2]=>4
[2,1,3,4,5]=>1
[2,1,3,5,4]=>2
[2,1,4,3,5]=>2
[2,1,4,5,3]=>3
[2,1,5,3,4]=>3
[2,1,5,4,3]=>3
[2,3,1,4,5]=>2
[2,3,1,5,4]=>3
[2,3,4,1,5]=>3
[2,3,4,5,1]=>4
[2,3,5,1,4]=>4
[2,3,5,4,1]=>4
[2,4,1,3,5]=>3
[2,4,1,5,3]=>4
[2,4,3,1,5]=>3
[2,4,3,5,1]=>4
[2,4,5,1,3]=>5
[2,4,5,3,1]=>5
[2,5,1,3,4]=>4
[2,5,1,4,3]=>4
[2,5,3,1,4]=>4
[2,5,3,4,1]=>4
[2,5,4,1,3]=>5
[2,5,4,3,1]=>5
[3,1,2,4,5]=>2
[3,1,2,5,4]=>3
[3,1,4,2,5]=>3
[3,1,4,5,2]=>4
[3,1,5,2,4]=>4
[3,1,5,4,2]=>4
[3,2,1,4,5]=>2
[3,2,1,5,4]=>3
[3,2,4,1,5]=>3
[3,2,4,5,1]=>4
[3,2,5,1,4]=>4
[3,2,5,4,1]=>4
[3,4,1,2,5]=>4
[3,4,1,5,2]=>5
[3,4,2,1,5]=>4
[3,4,2,5,1]=>5
[3,4,5,1,2]=>6
[3,4,5,2,1]=>6
[3,5,1,2,4]=>5
[3,5,1,4,2]=>5
[3,5,2,1,4]=>5
[3,5,2,4,1]=>5
[3,5,4,1,2]=>6
[3,5,4,2,1]=>6
[4,1,2,3,5]=>3
[4,1,2,5,3]=>4
[4,1,3,2,5]=>3
[4,1,3,5,2]=>4
[4,1,5,2,3]=>5
[4,1,5,3,2]=>5
[4,2,1,3,5]=>3
[4,2,1,5,3]=>4
[4,2,3,1,5]=>3
[4,2,3,5,1]=>4
[4,2,5,1,3]=>5
[4,2,5,3,1]=>5
[4,3,1,2,5]=>4
[4,3,1,5,2]=>5
[4,3,2,1,5]=>4
[4,3,2,5,1]=>5
[4,3,5,1,2]=>6
[4,3,5,2,1]=>6
[4,5,1,2,3]=>6
[4,5,1,3,2]=>6
[4,5,2,1,3]=>6
[4,5,2,3,1]=>6
[4,5,3,1,2]=>6
[4,5,3,2,1]=>6
[5,1,2,3,4]=>4
[5,1,2,4,3]=>4
[5,1,3,2,4]=>4
[5,1,3,4,2]=>4
[5,1,4,2,3]=>5
[5,1,4,3,2]=>5
[5,2,1,3,4]=>4
[5,2,1,4,3]=>4
[5,2,3,1,4]=>4
[5,2,3,4,1]=>4
[5,2,4,1,3]=>5
[5,2,4,3,1]=>5
[5,3,1,2,4]=>5
[5,3,1,4,2]=>5
[5,3,2,1,4]=>5
[5,3,2,4,1]=>5
[5,3,4,1,2]=>6
[5,3,4,2,1]=>6
[5,4,1,2,3]=>6
[5,4,1,3,2]=>6
[5,4,2,1,3]=>6
[5,4,2,3,1]=>6
[5,4,3,1,2]=>6
[5,4,3,2,1]=>6
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>1
[1,2,3,5,4,6]=>1
[1,2,3,5,6,4]=>2
[1,2,3,6,4,5]=>2
[1,2,3,6,5,4]=>2
[1,2,4,3,5,6]=>1
[1,2,4,3,6,5]=>2
[1,2,4,5,3,6]=>2
[1,2,4,5,6,3]=>3
[1,2,4,6,3,5]=>3
[1,2,4,6,5,3]=>3
[1,2,5,3,4,6]=>2
[1,2,5,3,6,4]=>3
[1,2,5,4,3,6]=>2
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>4
[1,2,5,6,4,3]=>4
[1,2,6,3,4,5]=>3
[1,2,6,3,5,4]=>3
[1,2,6,4,3,5]=>3
[1,2,6,4,5,3]=>3
[1,2,6,5,3,4]=>4
[1,2,6,5,4,3]=>4
[1,3,2,4,5,6]=>1
[1,3,2,4,6,5]=>2
[1,3,2,5,4,6]=>2
[1,3,2,5,6,4]=>3
[1,3,2,6,4,5]=>3
[1,3,2,6,5,4]=>3
[1,3,4,2,5,6]=>2
[1,3,4,2,6,5]=>3
[1,3,4,5,2,6]=>3
[1,3,4,5,6,2]=>4
[1,3,4,6,2,5]=>4
[1,3,4,6,5,2]=>4
[1,3,5,2,4,6]=>3
[1,3,5,2,6,4]=>4
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>4
[1,3,5,6,2,4]=>5
[1,3,5,6,4,2]=>5
[1,3,6,2,4,5]=>4
[1,3,6,2,5,4]=>4
[1,3,6,4,2,5]=>4
[1,3,6,4,5,2]=>4
[1,3,6,5,2,4]=>5
[1,3,6,5,4,2]=>5
[1,4,2,3,5,6]=>2
[1,4,2,3,6,5]=>3
[1,4,2,5,3,6]=>3
[1,4,2,5,6,3]=>4
[1,4,2,6,3,5]=>4
[1,4,2,6,5,3]=>4
[1,4,3,2,5,6]=>2
[1,4,3,2,6,5]=>3
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>4
[1,4,3,6,2,5]=>4
[1,4,3,6,5,2]=>4
[1,4,5,2,3,6]=>4
[1,4,5,2,6,3]=>5
[1,4,5,3,2,6]=>4
[1,4,5,3,6,2]=>5
[1,4,5,6,2,3]=>6
[1,4,5,6,3,2]=>6
[1,4,6,2,3,5]=>5
[1,4,6,2,5,3]=>5
[1,4,6,3,2,5]=>5
[1,4,6,3,5,2]=>5
[1,4,6,5,2,3]=>6
[1,4,6,5,3,2]=>6
[1,5,2,3,4,6]=>3
[1,5,2,3,6,4]=>4
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>4
[1,5,2,6,3,4]=>5
[1,5,2,6,4,3]=>5
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>4
[1,5,3,4,2,6]=>3
[1,5,3,4,6,2]=>4
[1,5,3,6,2,4]=>5
[1,5,3,6,4,2]=>5
[1,5,4,2,3,6]=>4
[1,5,4,2,6,3]=>5
[1,5,4,3,2,6]=>4
[1,5,4,3,6,2]=>5
[1,5,4,6,2,3]=>6
[1,5,4,6,3,2]=>6
[1,5,6,2,3,4]=>6
[1,5,6,2,4,3]=>6
[1,5,6,3,2,4]=>6
[1,5,6,3,4,2]=>6
[1,5,6,4,2,3]=>6
[1,5,6,4,3,2]=>6
[1,6,2,3,4,5]=>4
[1,6,2,3,5,4]=>4
[1,6,2,4,3,5]=>4
[1,6,2,4,5,3]=>4
[1,6,2,5,3,4]=>5
[1,6,2,5,4,3]=>5
[1,6,3,2,4,5]=>4
[1,6,3,2,5,4]=>4
[1,6,3,4,2,5]=>4
[1,6,3,4,5,2]=>4
[1,6,3,5,2,4]=>5
[1,6,3,5,4,2]=>5
[1,6,4,2,3,5]=>5
[1,6,4,2,5,3]=>5
[1,6,4,3,2,5]=>5
[1,6,4,3,5,2]=>5
[1,6,4,5,2,3]=>6
[1,6,4,5,3,2]=>6
[1,6,5,2,3,4]=>6
[1,6,5,2,4,3]=>6
[1,6,5,3,2,4]=>6
[1,6,5,3,4,2]=>6
[1,6,5,4,2,3]=>6
[1,6,5,4,3,2]=>6
[2,1,3,4,5,6]=>1
[2,1,3,4,6,5]=>2
[2,1,3,5,4,6]=>2
[2,1,3,5,6,4]=>3
[2,1,3,6,4,5]=>3
[2,1,3,6,5,4]=>3
[2,1,4,3,5,6]=>2
[2,1,4,3,6,5]=>3
[2,1,4,5,3,6]=>3
[2,1,4,5,6,3]=>4
[2,1,4,6,3,5]=>4
[2,1,4,6,5,3]=>4
[2,1,5,3,4,6]=>3
[2,1,5,3,6,4]=>4
[2,1,5,4,3,6]=>3
[2,1,5,4,6,3]=>4
[2,1,5,6,3,4]=>5
[2,1,5,6,4,3]=>5
[2,1,6,3,4,5]=>4
[2,1,6,3,5,4]=>4
[2,1,6,4,3,5]=>4
[2,1,6,4,5,3]=>4
[2,1,6,5,3,4]=>5
[2,1,6,5,4,3]=>5
[2,3,1,4,5,6]=>2
[2,3,1,4,6,5]=>3
[2,3,1,5,4,6]=>3
[2,3,1,5,6,4]=>4
[2,3,1,6,4,5]=>4
[2,3,1,6,5,4]=>4
[2,3,4,1,5,6]=>3
[2,3,4,1,6,5]=>4
[2,3,4,5,1,6]=>4
[2,3,4,5,6,1]=>5
[2,3,4,6,1,5]=>5
[2,3,4,6,5,1]=>5
[2,3,5,1,4,6]=>4
[2,3,5,1,6,4]=>5
[2,3,5,4,1,6]=>4
[2,3,5,4,6,1]=>5
[2,3,5,6,1,4]=>6
[2,3,5,6,4,1]=>6
[2,3,6,1,4,5]=>5
[2,3,6,1,5,4]=>5
[2,3,6,4,1,5]=>5
[2,3,6,4,5,1]=>5
[2,3,6,5,1,4]=>6
[2,3,6,5,4,1]=>6
[2,4,1,3,5,6]=>3
[2,4,1,3,6,5]=>4
[2,4,1,5,3,6]=>4
[2,4,1,5,6,3]=>5
[2,4,1,6,3,5]=>5
[2,4,1,6,5,3]=>5
[2,4,3,1,5,6]=>3
[2,4,3,1,6,5]=>4
[2,4,3,5,1,6]=>4
[2,4,3,5,6,1]=>5
[2,4,3,6,1,5]=>5
[2,4,3,6,5,1]=>5
[2,4,5,1,3,6]=>5
[2,4,5,1,6,3]=>6
[2,4,5,3,1,6]=>5
[2,4,5,3,6,1]=>6
[2,4,5,6,1,3]=>7
[2,4,5,6,3,1]=>7
[2,4,6,1,3,5]=>6
[2,4,6,1,5,3]=>6
[2,4,6,3,1,5]=>6
[2,4,6,3,5,1]=>6
[2,4,6,5,1,3]=>7
[2,4,6,5,3,1]=>7
[2,5,1,3,4,6]=>4
[2,5,1,3,6,4]=>5
[2,5,1,4,3,6]=>4
[2,5,1,4,6,3]=>5
[2,5,1,6,3,4]=>6
[2,5,1,6,4,3]=>6
[2,5,3,1,4,6]=>4
[2,5,3,1,6,4]=>5
[2,5,3,4,1,6]=>4
[2,5,3,4,6,1]=>5
[2,5,3,6,1,4]=>6
[2,5,3,6,4,1]=>6
[2,5,4,1,3,6]=>5
[2,5,4,1,6,3]=>6
[2,5,4,3,1,6]=>5
[2,5,4,3,6,1]=>6
[2,5,4,6,1,3]=>7
[2,5,4,6,3,1]=>7
[2,5,6,1,3,4]=>7
[2,5,6,1,4,3]=>7
[2,5,6,3,1,4]=>7
[2,5,6,3,4,1]=>7
[2,5,6,4,1,3]=>7
[2,5,6,4,3,1]=>7
[2,6,1,3,4,5]=>5
[2,6,1,3,5,4]=>5
[2,6,1,4,3,5]=>5
[2,6,1,4,5,3]=>5
[2,6,1,5,3,4]=>6
[2,6,1,5,4,3]=>6
[2,6,3,1,4,5]=>5
[2,6,3,1,5,4]=>5
[2,6,3,4,1,5]=>5
[2,6,3,4,5,1]=>5
[2,6,3,5,1,4]=>6
[2,6,3,5,4,1]=>6
[2,6,4,1,3,5]=>6
[2,6,4,1,5,3]=>6
[2,6,4,3,1,5]=>6
[2,6,4,3,5,1]=>6
[2,6,4,5,1,3]=>7
[2,6,4,5,3,1]=>7
[2,6,5,1,3,4]=>7
[2,6,5,1,4,3]=>7
[2,6,5,3,1,4]=>7
[2,6,5,3,4,1]=>7
[2,6,5,4,1,3]=>7
[2,6,5,4,3,1]=>7
[3,1,2,4,5,6]=>2
[3,1,2,4,6,5]=>3
[3,1,2,5,4,6]=>3
[3,1,2,5,6,4]=>4
[3,1,2,6,4,5]=>4
[3,1,2,6,5,4]=>4
[3,1,4,2,5,6]=>3
[3,1,4,2,6,5]=>4
[3,1,4,5,2,6]=>4
[3,1,4,5,6,2]=>5
[3,1,4,6,2,5]=>5
[3,1,4,6,5,2]=>5
[3,1,5,2,4,6]=>4
[3,1,5,2,6,4]=>5
[3,1,5,4,2,6]=>4
[3,1,5,4,6,2]=>5
[3,1,5,6,2,4]=>6
[3,1,5,6,4,2]=>6
[3,1,6,2,4,5]=>5
[3,1,6,2,5,4]=>5
[3,1,6,4,2,5]=>5
[3,1,6,4,5,2]=>5
[3,1,6,5,2,4]=>6
[3,1,6,5,4,2]=>6
[3,2,1,4,5,6]=>2
[3,2,1,4,6,5]=>3
[3,2,1,5,4,6]=>3
[3,2,1,5,6,4]=>4
[3,2,1,6,4,5]=>4
[3,2,1,6,5,4]=>4
[3,2,4,1,5,6]=>3
[3,2,4,1,6,5]=>4
[3,2,4,5,1,6]=>4
[3,2,4,5,6,1]=>5
[3,2,4,6,1,5]=>5
[3,2,4,6,5,1]=>5
[3,2,5,1,4,6]=>4
[3,2,5,1,6,4]=>5
[3,2,5,4,1,6]=>4
[3,2,5,4,6,1]=>5
[3,2,5,6,1,4]=>6
[3,2,5,6,4,1]=>6
[3,2,6,1,4,5]=>5
[3,2,6,1,5,4]=>5
[3,2,6,4,1,5]=>5
[3,2,6,4,5,1]=>5
[3,2,6,5,1,4]=>6
[3,2,6,5,4,1]=>6
[3,4,1,2,5,6]=>4
[3,4,1,2,6,5]=>5
[3,4,1,5,2,6]=>5
[3,4,1,5,6,2]=>6
[3,4,1,6,2,5]=>6
[3,4,1,6,5,2]=>6
[3,4,2,1,5,6]=>4
[3,4,2,1,6,5]=>5
[3,4,2,5,1,6]=>5
[3,4,2,5,6,1]=>6
[3,4,2,6,1,5]=>6
[3,4,2,6,5,1]=>6
[3,4,5,1,2,6]=>6
[3,4,5,1,6,2]=>7
[3,4,5,2,1,6]=>6
[3,4,5,2,6,1]=>7
[3,4,5,6,1,2]=>8
[3,4,5,6,2,1]=>8
[3,4,6,1,2,5]=>7
[3,4,6,1,5,2]=>7
[3,4,6,2,1,5]=>7
[3,4,6,2,5,1]=>7
[3,4,6,5,1,2]=>8
[3,4,6,5,2,1]=>8
[3,5,1,2,4,6]=>5
[3,5,1,2,6,4]=>6
[3,5,1,4,2,6]=>5
[3,5,1,4,6,2]=>6
[3,5,1,6,2,4]=>7
[3,5,1,6,4,2]=>7
[3,5,2,1,4,6]=>5
[3,5,2,1,6,4]=>6
[3,5,2,4,1,6]=>5
[3,5,2,4,6,1]=>6
[3,5,2,6,1,4]=>7
[3,5,2,6,4,1]=>7
[3,5,4,1,2,6]=>6
[3,5,4,1,6,2]=>7
[3,5,4,2,1,6]=>6
[3,5,4,2,6,1]=>7
[3,5,4,6,1,2]=>8
[3,5,4,6,2,1]=>8
[3,5,6,1,2,4]=>8
[3,5,6,1,4,2]=>8
[3,5,6,2,1,4]=>8
[3,5,6,2,4,1]=>8
[3,5,6,4,1,2]=>8
[3,5,6,4,2,1]=>8
[3,6,1,2,4,5]=>6
[3,6,1,2,5,4]=>6
[3,6,1,4,2,5]=>6
[3,6,1,4,5,2]=>6
[3,6,1,5,2,4]=>7
[3,6,1,5,4,2]=>7
[3,6,2,1,4,5]=>6
[3,6,2,1,5,4]=>6
[3,6,2,4,1,5]=>6
[3,6,2,4,5,1]=>6
[3,6,2,5,1,4]=>7
[3,6,2,5,4,1]=>7
[3,6,4,1,2,5]=>7
[3,6,4,1,5,2]=>7
[3,6,4,2,1,5]=>7
[3,6,4,2,5,1]=>7
[3,6,4,5,1,2]=>8
[3,6,4,5,2,1]=>8
[3,6,5,1,2,4]=>8
[3,6,5,1,4,2]=>8
[3,6,5,2,1,4]=>8
[3,6,5,2,4,1]=>8
[3,6,5,4,1,2]=>8
[3,6,5,4,2,1]=>8
[4,1,2,3,5,6]=>3
[4,1,2,3,6,5]=>4
[4,1,2,5,3,6]=>4
[4,1,2,5,6,3]=>5
[4,1,2,6,3,5]=>5
[4,1,2,6,5,3]=>5
[4,1,3,2,5,6]=>3
[4,1,3,2,6,5]=>4
[4,1,3,5,2,6]=>4
[4,1,3,5,6,2]=>5
[4,1,3,6,2,5]=>5
[4,1,3,6,5,2]=>5
[4,1,5,2,3,6]=>5
[4,1,5,2,6,3]=>6
[4,1,5,3,2,6]=>5
[4,1,5,3,6,2]=>6
[4,1,5,6,2,3]=>7
[4,1,5,6,3,2]=>7
[4,1,6,2,3,5]=>6
[4,1,6,2,5,3]=>6
[4,1,6,3,2,5]=>6
[4,1,6,3,5,2]=>6
[4,1,6,5,2,3]=>7
[4,1,6,5,3,2]=>7
[4,2,1,3,5,6]=>3
[4,2,1,3,6,5]=>4
[4,2,1,5,3,6]=>4
[4,2,1,5,6,3]=>5
[4,2,1,6,3,5]=>5
[4,2,1,6,5,3]=>5
[4,2,3,1,5,6]=>3
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>4
[4,2,3,5,6,1]=>5
[4,2,3,6,1,5]=>5
[4,2,3,6,5,1]=>5
[4,2,5,1,3,6]=>5
[4,2,5,1,6,3]=>6
[4,2,5,3,1,6]=>5
[4,2,5,3,6,1]=>6
[4,2,5,6,1,3]=>7
[4,2,5,6,3,1]=>7
[4,2,6,1,3,5]=>6
[4,2,6,1,5,3]=>6
[4,2,6,3,1,5]=>6
[4,2,6,3,5,1]=>6
[4,2,6,5,1,3]=>7
[4,2,6,5,3,1]=>7
[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]=>6
[4,3,1,6,2,5]=>6
[4,3,1,6,5,2]=>6
[4,3,2,1,5,6]=>4
[4,3,2,1,6,5]=>5
[4,3,2,5,1,6]=>5
[4,3,2,5,6,1]=>6
[4,3,2,6,1,5]=>6
[4,3,2,6,5,1]=>6
[4,3,5,1,2,6]=>6
[4,3,5,1,6,2]=>7
[4,3,5,2,1,6]=>6
[4,3,5,2,6,1]=>7
[4,3,5,6,1,2]=>8
[4,3,5,6,2,1]=>8
[4,3,6,1,2,5]=>7
[4,3,6,1,5,2]=>7
[4,3,6,2,1,5]=>7
[4,3,6,2,5,1]=>7
[4,3,6,5,1,2]=>8
[4,3,6,5,2,1]=>8
[4,5,1,2,3,6]=>6
[4,5,1,2,6,3]=>7
[4,5,1,3,2,6]=>6
[4,5,1,3,6,2]=>7
[4,5,1,6,2,3]=>8
[4,5,1,6,3,2]=>8
[4,5,2,1,3,6]=>6
[4,5,2,1,6,3]=>7
[4,5,2,3,1,6]=>6
[4,5,2,3,6,1]=>7
[4,5,2,6,1,3]=>8
[4,5,2,6,3,1]=>8
[4,5,3,1,2,6]=>6
[4,5,3,1,6,2]=>7
[4,5,3,2,1,6]=>6
[4,5,3,2,6,1]=>7
[4,5,3,6,1,2]=>8
[4,5,3,6,2,1]=>8
[4,5,6,1,2,3]=>9
[4,5,6,1,3,2]=>9
[4,5,6,2,1,3]=>9
[4,5,6,2,3,1]=>9
[4,5,6,3,1,2]=>9
[4,5,6,3,2,1]=>9
[4,6,1,2,3,5]=>7
[4,6,1,2,5,3]=>7
[4,6,1,3,2,5]=>7
[4,6,1,3,5,2]=>7
[4,6,1,5,2,3]=>8
[4,6,1,5,3,2]=>8
[4,6,2,1,3,5]=>7
[4,6,2,1,5,3]=>7
[4,6,2,3,1,5]=>7
[4,6,2,3,5,1]=>7
[4,6,2,5,1,3]=>8
[4,6,2,5,3,1]=>8
[4,6,3,1,2,5]=>7
[4,6,3,1,5,2]=>7
[4,6,3,2,1,5]=>7
[4,6,3,2,5,1]=>7
[4,6,3,5,1,2]=>8
[4,6,3,5,2,1]=>8
[4,6,5,1,2,3]=>9
[4,6,5,1,3,2]=>9
[4,6,5,2,1,3]=>9
[4,6,5,2,3,1]=>9
[4,6,5,3,1,2]=>9
[4,6,5,3,2,1]=>9
[5,1,2,3,4,6]=>4
[5,1,2,3,6,4]=>5
[5,1,2,4,3,6]=>4
[5,1,2,4,6,3]=>5
[5,1,2,6,3,4]=>6
[5,1,2,6,4,3]=>6
[5,1,3,2,4,6]=>4
[5,1,3,2,6,4]=>5
[5,1,3,4,2,6]=>4
[5,1,3,4,6,2]=>5
[5,1,3,6,2,4]=>6
[5,1,3,6,4,2]=>6
[5,1,4,2,3,6]=>5
[5,1,4,2,6,3]=>6
[5,1,4,3,2,6]=>5
[5,1,4,3,6,2]=>6
[5,1,4,6,2,3]=>7
[5,1,4,6,3,2]=>7
[5,1,6,2,3,4]=>7
[5,1,6,2,4,3]=>7
[5,1,6,3,2,4]=>7
[5,1,6,3,4,2]=>7
[5,1,6,4,2,3]=>7
[5,1,6,4,3,2]=>7
[5,2,1,3,4,6]=>4
[5,2,1,3,6,4]=>5
[5,2,1,4,3,6]=>4
[5,2,1,4,6,3]=>5
[5,2,1,6,3,4]=>6
[5,2,1,6,4,3]=>6
[5,2,3,1,4,6]=>4
[5,2,3,1,6,4]=>5
[5,2,3,4,1,6]=>4
[5,2,3,4,6,1]=>5
[5,2,3,6,1,4]=>6
[5,2,3,6,4,1]=>6
[5,2,4,1,3,6]=>5
[5,2,4,1,6,3]=>6
[5,2,4,3,1,6]=>5
[5,2,4,3,6,1]=>6
[5,2,4,6,1,3]=>7
[5,2,4,6,3,1]=>7
[5,2,6,1,3,4]=>7
[5,2,6,1,4,3]=>7
[5,2,6,3,1,4]=>7
[5,2,6,3,4,1]=>7
[5,2,6,4,1,3]=>7
[5,2,6,4,3,1]=>7
[5,3,1,2,4,6]=>5
[5,3,1,2,6,4]=>6
[5,3,1,4,2,6]=>5
[5,3,1,4,6,2]=>6
[5,3,1,6,2,4]=>7
[5,3,1,6,4,2]=>7
[5,3,2,1,4,6]=>5
[5,3,2,1,6,4]=>6
[5,3,2,4,1,6]=>5
[5,3,2,4,6,1]=>6
[5,3,2,6,1,4]=>7
[5,3,2,6,4,1]=>7
[5,3,4,1,2,6]=>6
[5,3,4,1,6,2]=>7
[5,3,4,2,1,6]=>6
[5,3,4,2,6,1]=>7
[5,3,4,6,1,2]=>8
[5,3,4,6,2,1]=>8
[5,3,6,1,2,4]=>8
[5,3,6,1,4,2]=>8
[5,3,6,2,1,4]=>8
[5,3,6,2,4,1]=>8
[5,3,6,4,1,2]=>8
[5,3,6,4,2,1]=>8
[5,4,1,2,3,6]=>6
[5,4,1,2,6,3]=>7
[5,4,1,3,2,6]=>6
[5,4,1,3,6,2]=>7
[5,4,1,6,2,3]=>8
[5,4,1,6,3,2]=>8
[5,4,2,1,3,6]=>6
[5,4,2,1,6,3]=>7
[5,4,2,3,1,6]=>6
[5,4,2,3,6,1]=>7
[5,4,2,6,1,3]=>8
[5,4,2,6,3,1]=>8
[5,4,3,1,2,6]=>6
[5,4,3,1,6,2]=>7
[5,4,3,2,1,6]=>6
[5,4,3,2,6,1]=>7
[5,4,3,6,1,2]=>8
[5,4,3,6,2,1]=>8
[5,4,6,1,2,3]=>9
[5,4,6,1,3,2]=>9
[5,4,6,2,1,3]=>9
[5,4,6,2,3,1]=>9
[5,4,6,3,1,2]=>9
[5,4,6,3,2,1]=>9
[5,6,1,2,3,4]=>8
[5,6,1,2,4,3]=>8
[5,6,1,3,2,4]=>8
[5,6,1,3,4,2]=>8
[5,6,1,4,2,3]=>8
[5,6,1,4,3,2]=>8
[5,6,2,1,3,4]=>8
[5,6,2,1,4,3]=>8
[5,6,2,3,1,4]=>8
[5,6,2,3,4,1]=>8
[5,6,2,4,1,3]=>8
[5,6,2,4,3,1]=>8
[5,6,3,1,2,4]=>8
[5,6,3,1,4,2]=>8
[5,6,3,2,1,4]=>8
[5,6,3,2,4,1]=>8
[5,6,3,4,1,2]=>8
[5,6,3,4,2,1]=>8
[5,6,4,1,2,3]=>9
[5,6,4,1,3,2]=>9
[5,6,4,2,1,3]=>9
[5,6,4,2,3,1]=>9
[5,6,4,3,1,2]=>9
[5,6,4,3,2,1]=>9
[6,1,2,3,4,5]=>5
[6,1,2,3,5,4]=>5
[6,1,2,4,3,5]=>5
[6,1,2,4,5,3]=>5
[6,1,2,5,3,4]=>6
[6,1,2,5,4,3]=>6
[6,1,3,2,4,5]=>5
[6,1,3,2,5,4]=>5
[6,1,3,4,2,5]=>5
[6,1,3,4,5,2]=>5
[6,1,3,5,2,4]=>6
[6,1,3,5,4,2]=>6
[6,1,4,2,3,5]=>6
[6,1,4,2,5,3]=>6
[6,1,4,3,2,5]=>6
[6,1,4,3,5,2]=>6
[6,1,4,5,2,3]=>7
[6,1,4,5,3,2]=>7
[6,1,5,2,3,4]=>7
[6,1,5,2,4,3]=>7
[6,1,5,3,2,4]=>7
[6,1,5,3,4,2]=>7
[6,1,5,4,2,3]=>7
[6,1,5,4,3,2]=>7
[6,2,1,3,4,5]=>5
[6,2,1,3,5,4]=>5
[6,2,1,4,3,5]=>5
[6,2,1,4,5,3]=>5
[6,2,1,5,3,4]=>6
[6,2,1,5,4,3]=>6
[6,2,3,1,4,5]=>5
[6,2,3,1,5,4]=>5
[6,2,3,4,1,5]=>5
[6,2,3,4,5,1]=>5
[6,2,3,5,1,4]=>6
[6,2,3,5,4,1]=>6
[6,2,4,1,3,5]=>6
[6,2,4,1,5,3]=>6
[6,2,4,3,1,5]=>6
[6,2,4,3,5,1]=>6
[6,2,4,5,1,3]=>7
[6,2,4,5,3,1]=>7
[6,2,5,1,3,4]=>7
[6,2,5,1,4,3]=>7
[6,2,5,3,1,4]=>7
[6,2,5,3,4,1]=>7
[6,2,5,4,1,3]=>7
[6,2,5,4,3,1]=>7
[6,3,1,2,4,5]=>6
[6,3,1,2,5,4]=>6
[6,3,1,4,2,5]=>6
[6,3,1,4,5,2]=>6
[6,3,1,5,2,4]=>7
[6,3,1,5,4,2]=>7
[6,3,2,1,4,5]=>6
[6,3,2,1,5,4]=>6
[6,3,2,4,1,5]=>6
[6,3,2,4,5,1]=>6
[6,3,2,5,1,4]=>7
[6,3,2,5,4,1]=>7
[6,3,4,1,2,5]=>7
[6,3,4,1,5,2]=>7
[6,3,4,2,1,5]=>7
[6,3,4,2,5,1]=>7
[6,3,4,5,1,2]=>8
[6,3,4,5,2,1]=>8
[6,3,5,1,2,4]=>8
[6,3,5,1,4,2]=>8
[6,3,5,2,1,4]=>8
[6,3,5,2,4,1]=>8
[6,3,5,4,1,2]=>8
[6,3,5,4,2,1]=>8
[6,4,1,2,3,5]=>7
[6,4,1,2,5,3]=>7
[6,4,1,3,2,5]=>7
[6,4,1,3,5,2]=>7
[6,4,1,5,2,3]=>8
[6,4,1,5,3,2]=>8
[6,4,2,1,3,5]=>7
[6,4,2,1,5,3]=>7
[6,4,2,3,1,5]=>7
[6,4,2,3,5,1]=>7
[6,4,2,5,1,3]=>8
[6,4,2,5,3,1]=>8
[6,4,3,1,2,5]=>7
[6,4,3,1,5,2]=>7
[6,4,3,2,1,5]=>7
[6,4,3,2,5,1]=>7
[6,4,3,5,1,2]=>8
[6,4,3,5,2,1]=>8
[6,4,5,1,2,3]=>9
[6,4,5,1,3,2]=>9
[6,4,5,2,1,3]=>9
[6,4,5,2,3,1]=>9
[6,4,5,3,1,2]=>9
[6,4,5,3,2,1]=>9
[6,5,1,2,3,4]=>8
[6,5,1,2,4,3]=>8
[6,5,1,3,2,4]=>8
[6,5,1,3,4,2]=>8
[6,5,1,4,2,3]=>8
[6,5,1,4,3,2]=>8
[6,5,2,1,3,4]=>8
[6,5,2,1,4,3]=>8
[6,5,2,3,1,4]=>8
[6,5,2,3,4,1]=>8
[6,5,2,4,1,3]=>8
[6,5,2,4,3,1]=>8
[6,5,3,1,2,4]=>8
[6,5,3,1,4,2]=>8
[6,5,3,2,1,4]=>8
[6,5,3,2,4,1]=>8
[6,5,3,4,1,2]=>8
[6,5,3,4,2,1]=>8
[6,5,4,1,2,3]=>9
[6,5,4,1,3,2]=>9
[6,5,4,2,1,3]=>9
[6,5,4,2,3,1]=>9
[6,5,4,3,1,2]=>9
[6,5,4,3,2,1]=>9
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Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called almost-increasing in [5].
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called almost-increasing in [5].
References
[1] Petersen, T. K., Tenner, B. E. The depth of a permutation arXiv:1202.4765
[2] Bagno, E., Biagioli, R., Novick, M., Woo, A. Depth in classical Coexter groups arXiv:1507.01180
[3] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray MathSciNet:0652736
[4] Knuth, D. E. The art of computer programming. Vol. 3 MathSciNet:3077154
[5] Elizalde, S. The $X$-class and almost-increasing permutations arXiv:0710.5168
[2] Bagno, E., Biagioli, R., Novick, M., Woo, A. Depth in classical Coexter groups arXiv:1507.01180
[3] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray MathSciNet:0652736
[4] Knuth, D. E. The art of computer programming. Vol. 3 MathSciNet:3077154
[5] Elizalde, S. The $X$-class and almost-increasing permutations arXiv:0710.5168
Code
def statistic(w):
return sum(w[i]-i-1 for i in range(len(w)) if w[i]>i+1)
Created
Nov 30, 2012 at 21:09 by Chris Berg
Updated
Jan 15, 2017 at 10:37 by Christian Stump
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