Identifier
- St001745: Permutations ⟶ ℤ
Values
=>
[1]=>0
[1,2]=>0
[2,1]=>0
[1,2,3]=>0
[1,3,2]=>0
[2,1,3]=>1
[2,3,1]=>0
[3,1,2]=>0
[3,2,1]=>0
[1,2,3,4]=>0
[1,2,4,3]=>0
[1,3,2,4]=>1
[1,3,4,2]=>0
[1,4,2,3]=>0
[1,4,3,2]=>0
[2,1,3,4]=>2
[2,1,4,3]=>2
[2,3,1,4]=>1
[2,3,4,1]=>0
[2,4,1,3]=>0
[2,4,3,1]=>0
[3,1,2,4]=>2
[3,1,4,2]=>1
[3,2,1,4]=>1
[3,2,4,1]=>1
[3,4,1,2]=>0
[3,4,2,1]=>0
[4,1,2,3]=>1
[4,1,3,2]=>0
[4,2,1,3]=>0
[4,2,3,1]=>0
[4,3,1,2]=>0
[4,3,2,1]=>0
[1,2,3,4,5]=>0
[1,2,3,5,4]=>0
[1,2,4,3,5]=>1
[1,2,4,5,3]=>0
[1,2,5,3,4]=>0
[1,2,5,4,3]=>0
[1,3,2,4,5]=>2
[1,3,2,5,4]=>2
[1,3,4,2,5]=>1
[1,3,4,5,2]=>0
[1,3,5,2,4]=>0
[1,3,5,4,2]=>0
[1,4,2,3,5]=>2
[1,4,2,5,3]=>1
[1,4,3,2,5]=>1
[1,4,3,5,2]=>1
[1,4,5,2,3]=>0
[1,4,5,3,2]=>0
[1,5,2,3,4]=>1
[1,5,2,4,3]=>0
[1,5,3,2,4]=>0
[1,5,3,4,2]=>0
[1,5,4,2,3]=>0
[1,5,4,3,2]=>0
[2,1,3,4,5]=>3
[2,1,3,5,4]=>3
[2,1,4,3,5]=>4
[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]=>2
[2,3,4,1,5]=>1
[2,3,4,5,1]=>0
[2,3,5,1,4]=>0
[2,3,5,4,1]=>0
[2,4,1,3,5]=>2
[2,4,1,5,3]=>1
[2,4,3,1,5]=>1
[2,4,3,5,1]=>1
[2,4,5,1,3]=>0
[2,4,5,3,1]=>0
[2,5,1,3,4]=>1
[2,5,1,4,3]=>0
[2,5,3,1,4]=>0
[2,5,3,4,1]=>0
[2,5,4,1,3]=>0
[2,5,4,3,1]=>0
[3,1,2,4,5]=>4
[3,1,2,5,4]=>4
[3,1,4,2,5]=>3
[3,1,4,5,2]=>2
[3,1,5,2,4]=>2
[3,1,5,4,2]=>2
[3,2,1,4,5]=>2
[3,2,1,5,4]=>2
[3,2,4,1,5]=>3
[3,2,4,5,1]=>2
[3,2,5,1,4]=>2
[3,2,5,4,1]=>2
[3,4,1,2,5]=>2
[3,4,1,5,2]=>1
[3,4,2,1,5]=>1
[3,4,2,5,1]=>1
[3,4,5,1,2]=>0
[3,4,5,2,1]=>0
[3,5,1,2,4]=>1
[3,5,1,4,2]=>0
[3,5,2,1,4]=>0
[3,5,2,4,1]=>0
[3,5,4,1,2]=>0
[3,5,4,2,1]=>0
[4,1,2,3,5]=>4
[4,1,2,5,3]=>3
[4,1,3,2,5]=>2
[4,1,3,5,2]=>2
[4,1,5,2,3]=>1
[4,1,5,3,2]=>1
[4,2,1,3,5]=>2
[4,2,1,5,3]=>1
[4,2,3,1,5]=>2
[4,2,3,5,1]=>2
[4,2,5,1,3]=>1
[4,2,5,3,1]=>1
[4,3,1,2,5]=>2
[4,3,1,5,2]=>1
[4,3,2,1,5]=>1
[4,3,2,5,1]=>1
[4,3,5,1,2]=>1
[4,3,5,2,1]=>1
[4,5,1,2,3]=>1
[4,5,1,3,2]=>0
[4,5,2,1,3]=>0
[4,5,2,3,1]=>0
[4,5,3,1,2]=>0
[4,5,3,2,1]=>0
[5,1,2,3,4]=>3
[5,1,2,4,3]=>2
[5,1,3,2,4]=>1
[5,1,3,4,2]=>1
[5,1,4,2,3]=>0
[5,1,4,3,2]=>0
[5,2,1,3,4]=>1
[5,2,1,4,3]=>0
[5,2,3,1,4]=>1
[5,2,3,4,1]=>1
[5,2,4,1,3]=>0
[5,2,4,3,1]=>0
[5,3,1,2,4]=>1
[5,3,1,4,2]=>0
[5,3,2,1,4]=>0
[5,3,2,4,1]=>0
[5,3,4,1,2]=>0
[5,3,4,2,1]=>0
[5,4,1,2,3]=>1
[5,4,1,3,2]=>0
[5,4,2,1,3]=>0
[5,4,2,3,1]=>0
[5,4,3,1,2]=>0
[5,4,3,2,1]=>0
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>0
[1,2,3,5,4,6]=>1
[1,2,3,5,6,4]=>0
[1,2,3,6,4,5]=>0
[1,2,3,6,5,4]=>0
[1,2,4,3,5,6]=>2
[1,2,4,3,6,5]=>2
[1,2,4,5,3,6]=>1
[1,2,4,5,6,3]=>0
[1,2,4,6,3,5]=>0
[1,2,4,6,5,3]=>0
[1,2,5,3,4,6]=>2
[1,2,5,3,6,4]=>1
[1,2,5,4,3,6]=>1
[1,2,5,4,6,3]=>1
[1,2,5,6,3,4]=>0
[1,2,5,6,4,3]=>0
[1,2,6,3,4,5]=>1
[1,2,6,3,5,4]=>0
[1,2,6,4,3,5]=>0
[1,2,6,4,5,3]=>0
[1,2,6,5,3,4]=>0
[1,2,6,5,4,3]=>0
[1,3,2,4,5,6]=>3
[1,3,2,4,6,5]=>3
[1,3,2,5,4,6]=>4
[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]=>2
[1,3,4,5,2,6]=>1
[1,3,4,5,6,2]=>0
[1,3,4,6,2,5]=>0
[1,3,4,6,5,2]=>0
[1,3,5,2,4,6]=>2
[1,3,5,2,6,4]=>1
[1,3,5,4,2,6]=>1
[1,3,5,4,6,2]=>1
[1,3,5,6,2,4]=>0
[1,3,5,6,4,2]=>0
[1,3,6,2,4,5]=>1
[1,3,6,2,5,4]=>0
[1,3,6,4,2,5]=>0
[1,3,6,4,5,2]=>0
[1,3,6,5,2,4]=>0
[1,3,6,5,4,2]=>0
[1,4,2,3,5,6]=>4
[1,4,2,3,6,5]=>4
[1,4,2,5,3,6]=>3
[1,4,2,5,6,3]=>2
[1,4,2,6,3,5]=>2
[1,4,2,6,5,3]=>2
[1,4,3,2,5,6]=>2
[1,4,3,2,6,5]=>2
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>2
[1,4,3,6,2,5]=>2
[1,4,3,6,5,2]=>2
[1,4,5,2,3,6]=>2
[1,4,5,2,6,3]=>1
[1,4,5,3,2,6]=>1
[1,4,5,3,6,2]=>1
[1,4,5,6,2,3]=>0
[1,4,5,6,3,2]=>0
[1,4,6,2,3,5]=>1
[1,4,6,2,5,3]=>0
[1,4,6,3,2,5]=>0
[1,4,6,3,5,2]=>0
[1,4,6,5,2,3]=>0
[1,4,6,5,3,2]=>0
[1,5,2,3,4,6]=>4
[1,5,2,3,6,4]=>3
[1,5,2,4,3,6]=>2
[1,5,2,4,6,3]=>2
[1,5,2,6,3,4]=>1
[1,5,2,6,4,3]=>1
[1,5,3,2,4,6]=>2
[1,5,3,2,6,4]=>1
[1,5,3,4,2,6]=>2
[1,5,3,4,6,2]=>2
[1,5,3,6,2,4]=>1
[1,5,3,6,4,2]=>1
[1,5,4,2,3,6]=>2
[1,5,4,2,6,3]=>1
[1,5,4,3,2,6]=>1
[1,5,4,3,6,2]=>1
[1,5,4,6,2,3]=>1
[1,5,4,6,3,2]=>1
[1,5,6,2,3,4]=>1
[1,5,6,2,4,3]=>0
[1,5,6,3,2,4]=>0
[1,5,6,3,4,2]=>0
[1,5,6,4,2,3]=>0
[1,5,6,4,3,2]=>0
[1,6,2,3,4,5]=>3
[1,6,2,3,5,4]=>2
[1,6,2,4,3,5]=>1
[1,6,2,4,5,3]=>1
[1,6,2,5,3,4]=>0
[1,6,2,5,4,3]=>0
[1,6,3,2,4,5]=>1
[1,6,3,2,5,4]=>0
[1,6,3,4,2,5]=>1
[1,6,3,4,5,2]=>1
[1,6,3,5,2,4]=>0
[1,6,3,5,4,2]=>0
[1,6,4,2,3,5]=>1
[1,6,4,2,5,3]=>0
[1,6,4,3,2,5]=>0
[1,6,4,3,5,2]=>0
[1,6,4,5,2,3]=>0
[1,6,4,5,3,2]=>0
[1,6,5,2,3,4]=>1
[1,6,5,2,4,3]=>0
[1,6,5,3,2,4]=>0
[1,6,5,3,4,2]=>0
[1,6,5,4,2,3]=>0
[1,6,5,4,3,2]=>0
[2,1,3,4,5,6]=>4
[2,1,3,4,6,5]=>4
[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]=>4
[2,1,4,3,5,6]=>6
[2,1,4,3,6,5]=>6
[2,1,4,5,3,6]=>5
[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]=>6
[2,1,5,3,6,4]=>5
[2,1,5,4,3,6]=>5
[2,1,5,4,6,3]=>5
[2,1,5,6,3,4]=>4
[2,1,5,6,4,3]=>4
[2,1,6,3,4,5]=>5
[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]=>4
[2,1,6,5,4,3]=>4
[2,3,1,4,5,6]=>3
[2,3,1,4,6,5]=>3
[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]=>3
[2,3,4,1,5,6]=>2
[2,3,4,1,6,5]=>2
[2,3,4,5,1,6]=>1
[2,3,4,5,6,1]=>0
[2,3,4,6,1,5]=>0
[2,3,4,6,5,1]=>0
[2,3,5,1,4,6]=>2
[2,3,5,1,6,4]=>1
[2,3,5,4,1,6]=>1
[2,3,5,4,6,1]=>1
[2,3,5,6,1,4]=>0
[2,3,5,6,4,1]=>0
[2,3,6,1,4,5]=>1
[2,3,6,1,5,4]=>0
[2,3,6,4,1,5]=>0
[2,3,6,4,5,1]=>0
[2,3,6,5,1,4]=>0
[2,3,6,5,4,1]=>0
[2,4,1,3,5,6]=>4
[2,4,1,3,6,5]=>4
[2,4,1,5,3,6]=>3
[2,4,1,5,6,3]=>2
[2,4,1,6,3,5]=>2
[2,4,1,6,5,3]=>2
[2,4,3,1,5,6]=>2
[2,4,3,1,6,5]=>2
[2,4,3,5,1,6]=>3
[2,4,3,5,6,1]=>2
[2,4,3,6,1,5]=>2
[2,4,3,6,5,1]=>2
[2,4,5,1,3,6]=>2
[2,4,5,1,6,3]=>1
[2,4,5,3,1,6]=>1
[2,4,5,3,6,1]=>1
[2,4,5,6,1,3]=>0
[2,4,5,6,3,1]=>0
[2,4,6,1,3,5]=>1
[2,4,6,1,5,3]=>0
[2,4,6,3,1,5]=>0
[2,4,6,3,5,1]=>0
[2,4,6,5,1,3]=>0
[2,4,6,5,3,1]=>0
[2,5,1,3,4,6]=>4
[2,5,1,3,6,4]=>3
[2,5,1,4,3,6]=>2
[2,5,1,4,6,3]=>2
[2,5,1,6,3,4]=>1
[2,5,1,6,4,3]=>1
[2,5,3,1,4,6]=>2
[2,5,3,1,6,4]=>1
[2,5,3,4,1,6]=>2
[2,5,3,4,6,1]=>2
[2,5,3,6,1,4]=>1
[2,5,3,6,4,1]=>1
[2,5,4,1,3,6]=>2
[2,5,4,1,6,3]=>1
[2,5,4,3,1,6]=>1
[2,5,4,3,6,1]=>1
[2,5,4,6,1,3]=>1
[2,5,4,6,3,1]=>1
[2,5,6,1,3,4]=>1
[2,5,6,1,4,3]=>0
[2,5,6,3,1,4]=>0
[2,5,6,3,4,1]=>0
[2,5,6,4,1,3]=>0
[2,5,6,4,3,1]=>0
[2,6,1,3,4,5]=>3
[2,6,1,3,5,4]=>2
[2,6,1,4,3,5]=>1
[2,6,1,4,5,3]=>1
[2,6,1,5,3,4]=>0
[2,6,1,5,4,3]=>0
[2,6,3,1,4,5]=>1
[2,6,3,1,5,4]=>0
[2,6,3,4,1,5]=>1
[2,6,3,4,5,1]=>1
[2,6,3,5,1,4]=>0
[2,6,3,5,4,1]=>0
[2,6,4,1,3,5]=>1
[2,6,4,1,5,3]=>0
[2,6,4,3,1,5]=>0
[2,6,4,3,5,1]=>0
[2,6,4,5,1,3]=>0
[2,6,4,5,3,1]=>0
[2,6,5,1,3,4]=>1
[2,6,5,1,4,3]=>0
[2,6,5,3,1,4]=>0
[2,6,5,3,4,1]=>0
[2,6,5,4,1,3]=>0
[2,6,5,4,3,1]=>0
[3,1,2,4,5,6]=>6
[3,1,2,4,6,5]=>6
[3,1,2,5,4,6]=>7
[3,1,2,5,6,4]=>6
[3,1,2,6,4,5]=>6
[3,1,2,6,5,4]=>6
[3,1,4,2,5,6]=>5
[3,1,4,2,6,5]=>5
[3,1,4,5,2,6]=>4
[3,1,4,5,6,2]=>3
[3,1,4,6,2,5]=>3
[3,1,4,6,5,2]=>3
[3,1,5,2,4,6]=>5
[3,1,5,2,6,4]=>4
[3,1,5,4,2,6]=>4
[3,1,5,4,6,2]=>4
[3,1,5,6,2,4]=>3
[3,1,5,6,4,2]=>3
[3,1,6,2,4,5]=>4
[3,1,6,2,5,4]=>3
[3,1,6,4,2,5]=>3
[3,1,6,4,5,2]=>3
[3,1,6,5,2,4]=>3
[3,1,6,5,4,2]=>3
[3,2,1,4,5,6]=>3
[3,2,1,4,6,5]=>3
[3,2,1,5,4,6]=>4
[3,2,1,5,6,4]=>3
[3,2,1,6,4,5]=>3
[3,2,1,6,5,4]=>3
[3,2,4,1,5,6]=>5
[3,2,4,1,6,5]=>5
[3,2,4,5,1,6]=>4
[3,2,4,5,6,1]=>3
[3,2,4,6,1,5]=>3
[3,2,4,6,5,1]=>3
[3,2,5,1,4,6]=>5
[3,2,5,1,6,4]=>4
[3,2,5,4,1,6]=>4
[3,2,5,4,6,1]=>4
[3,2,5,6,1,4]=>3
[3,2,5,6,4,1]=>3
[3,2,6,1,4,5]=>4
[3,2,6,1,5,4]=>3
[3,2,6,4,1,5]=>3
[3,2,6,4,5,1]=>3
[3,2,6,5,1,4]=>3
[3,2,6,5,4,1]=>3
[3,4,1,2,5,6]=>4
[3,4,1,2,6,5]=>4
[3,4,1,5,2,6]=>3
[3,4,1,5,6,2]=>2
[3,4,1,6,2,5]=>2
[3,4,1,6,5,2]=>2
[3,4,2,1,5,6]=>2
[3,4,2,1,6,5]=>2
[3,4,2,5,1,6]=>3
[3,4,2,5,6,1]=>2
[3,4,2,6,1,5]=>2
[3,4,2,6,5,1]=>2
[3,4,5,1,2,6]=>2
[3,4,5,1,6,2]=>1
[3,4,5,2,1,6]=>1
[3,4,5,2,6,1]=>1
[3,4,5,6,1,2]=>0
[3,4,5,6,2,1]=>0
[3,4,6,1,2,5]=>1
[3,4,6,1,5,2]=>0
[3,4,6,2,1,5]=>0
[3,4,6,2,5,1]=>0
[3,4,6,5,1,2]=>0
[3,4,6,5,2,1]=>0
[3,5,1,2,4,6]=>4
[3,5,1,2,6,4]=>3
[3,5,1,4,2,6]=>2
[3,5,1,4,6,2]=>2
[3,5,1,6,2,4]=>1
[3,5,1,6,4,2]=>1
[3,5,2,1,4,6]=>2
[3,5,2,1,6,4]=>1
[3,5,2,4,1,6]=>2
[3,5,2,4,6,1]=>2
[3,5,2,6,1,4]=>1
[3,5,2,6,4,1]=>1
[3,5,4,1,2,6]=>2
[3,5,4,1,6,2]=>1
[3,5,4,2,1,6]=>1
[3,5,4,2,6,1]=>1
[3,5,4,6,1,2]=>1
[3,5,4,6,2,1]=>1
[3,5,6,1,2,4]=>1
[3,5,6,1,4,2]=>0
[3,5,6,2,1,4]=>0
[3,5,6,2,4,1]=>0
[3,5,6,4,1,2]=>0
[3,5,6,4,2,1]=>0
[3,6,1,2,4,5]=>3
[3,6,1,2,5,4]=>2
[3,6,1,4,2,5]=>1
[3,6,1,4,5,2]=>1
[3,6,1,5,2,4]=>0
[3,6,1,5,4,2]=>0
[3,6,2,1,4,5]=>1
[3,6,2,1,5,4]=>0
[3,6,2,4,1,5]=>1
[3,6,2,4,5,1]=>1
[3,6,2,5,1,4]=>0
[3,6,2,5,4,1]=>0
[3,6,4,1,2,5]=>1
[3,6,4,1,5,2]=>0
[3,6,4,2,1,5]=>0
[3,6,4,2,5,1]=>0
[3,6,4,5,1,2]=>0
[3,6,4,5,2,1]=>0
[3,6,5,1,2,4]=>1
[3,6,5,1,4,2]=>0
[3,6,5,2,1,4]=>0
[3,6,5,2,4,1]=>0
[3,6,5,4,1,2]=>0
[3,6,5,4,2,1]=>0
[4,1,2,3,5,6]=>7
[4,1,2,3,6,5]=>7
[4,1,2,5,3,6]=>6
[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]=>4
[4,1,3,2,6,5]=>4
[4,1,3,5,2,6]=>5
[4,1,3,5,6,2]=>4
[4,1,3,6,2,5]=>4
[4,1,3,6,5,2]=>4
[4,1,5,2,3,6]=>4
[4,1,5,2,6,3]=>3
[4,1,5,3,2,6]=>3
[4,1,5,3,6,2]=>3
[4,1,5,6,2,3]=>2
[4,1,5,6,3,2]=>2
[4,1,6,2,3,5]=>3
[4,1,6,2,5,3]=>2
[4,1,6,3,2,5]=>2
[4,1,6,3,5,2]=>2
[4,1,6,5,2,3]=>2
[4,1,6,5,3,2]=>2
[4,2,1,3,5,6]=>4
[4,2,1,3,6,5]=>4
[4,2,1,5,3,6]=>3
[4,2,1,5,6,3]=>2
[4,2,1,6,3,5]=>2
[4,2,1,6,5,3]=>2
[4,2,3,1,5,6]=>4
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>5
[4,2,3,5,6,1]=>4
[4,2,3,6,1,5]=>4
[4,2,3,6,5,1]=>4
[4,2,5,1,3,6]=>4
[4,2,5,1,6,3]=>3
[4,2,5,3,1,6]=>3
[4,2,5,3,6,1]=>3
[4,2,5,6,1,3]=>2
[4,2,5,6,3,1]=>2
[4,2,6,1,3,5]=>3
[4,2,6,1,5,3]=>2
[4,2,6,3,1,5]=>2
[4,2,6,3,5,1]=>2
[4,2,6,5,1,3]=>2
[4,2,6,5,3,1]=>2
[4,3,1,2,5,6]=>4
[4,3,1,2,6,5]=>4
[4,3,1,5,2,6]=>3
[4,3,1,5,6,2]=>2
[4,3,1,6,2,5]=>2
[4,3,1,6,5,2]=>2
[4,3,2,1,5,6]=>2
[4,3,2,1,6,5]=>2
[4,3,2,5,1,6]=>3
[4,3,2,5,6,1]=>2
[4,3,2,6,1,5]=>2
[4,3,2,6,5,1]=>2
[4,3,5,1,2,6]=>4
[4,3,5,1,6,2]=>3
[4,3,5,2,1,6]=>3
[4,3,5,2,6,1]=>3
[4,3,5,6,1,2]=>2
[4,3,5,6,2,1]=>2
[4,3,6,1,2,5]=>3
[4,3,6,1,5,2]=>2
[4,3,6,2,1,5]=>2
[4,3,6,2,5,1]=>2
[4,3,6,5,1,2]=>2
[4,3,6,5,2,1]=>2
[4,5,1,2,3,6]=>4
[4,5,1,2,6,3]=>3
[4,5,1,3,2,6]=>2
[4,5,1,3,6,2]=>2
[4,5,1,6,2,3]=>1
[4,5,1,6,3,2]=>1
[4,5,2,1,3,6]=>2
[4,5,2,1,6,3]=>1
[4,5,2,3,1,6]=>2
[4,5,2,3,6,1]=>2
[4,5,2,6,1,3]=>1
[4,5,2,6,3,1]=>1
[4,5,3,1,2,6]=>2
[4,5,3,1,6,2]=>1
[4,5,3,2,1,6]=>1
[4,5,3,2,6,1]=>1
[4,5,3,6,1,2]=>1
[4,5,3,6,2,1]=>1
[4,5,6,1,2,3]=>1
[4,5,6,1,3,2]=>0
[4,5,6,2,1,3]=>0
[4,5,6,2,3,1]=>0
[4,5,6,3,1,2]=>0
[4,5,6,3,2,1]=>0
[4,6,1,2,3,5]=>3
[4,6,1,2,5,3]=>2
[4,6,1,3,2,5]=>1
[4,6,1,3,5,2]=>1
[4,6,1,5,2,3]=>0
[4,6,1,5,3,2]=>0
[4,6,2,1,3,5]=>1
[4,6,2,1,5,3]=>0
[4,6,2,3,1,5]=>1
[4,6,2,3,5,1]=>1
[4,6,2,5,1,3]=>0
[4,6,2,5,3,1]=>0
[4,6,3,1,2,5]=>1
[4,6,3,1,5,2]=>0
[4,6,3,2,1,5]=>0
[4,6,3,2,5,1]=>0
[4,6,3,5,1,2]=>0
[4,6,3,5,2,1]=>0
[4,6,5,1,2,3]=>1
[4,6,5,1,3,2]=>0
[4,6,5,2,1,3]=>0
[4,6,5,2,3,1]=>0
[4,6,5,3,1,2]=>0
[4,6,5,3,2,1]=>0
[5,1,2,3,4,6]=>7
[5,1,2,3,6,4]=>6
[5,1,2,4,3,6]=>5
[5,1,2,4,6,3]=>5
[5,1,2,6,3,4]=>4
[5,1,2,6,4,3]=>4
[5,1,3,2,4,6]=>4
[5,1,3,2,6,4]=>3
[5,1,3,4,2,6]=>4
[5,1,3,4,6,2]=>4
[5,1,3,6,2,4]=>3
[5,1,3,6,4,2]=>3
[5,1,4,2,3,6]=>3
[5,1,4,2,6,3]=>2
[5,1,4,3,2,6]=>2
[5,1,4,3,6,2]=>2
[5,1,4,6,2,3]=>2
[5,1,4,6,3,2]=>2
[5,1,6,2,3,4]=>2
[5,1,6,2,4,3]=>1
[5,1,6,3,2,4]=>1
[5,1,6,3,4,2]=>1
[5,1,6,4,2,3]=>1
[5,1,6,4,3,2]=>1
[5,2,1,3,4,6]=>4
[5,2,1,3,6,4]=>3
[5,2,1,4,3,6]=>2
[5,2,1,4,6,3]=>2
[5,2,1,6,3,4]=>1
[5,2,1,6,4,3]=>1
[5,2,3,1,4,6]=>4
[5,2,3,1,6,4]=>3
[5,2,3,4,1,6]=>4
[5,2,3,4,6,1]=>4
[5,2,3,6,1,4]=>3
[5,2,3,6,4,1]=>3
[5,2,4,1,3,6]=>3
[5,2,4,1,6,3]=>2
[5,2,4,3,1,6]=>2
[5,2,4,3,6,1]=>2
[5,2,4,6,1,3]=>2
[5,2,4,6,3,1]=>2
[5,2,6,1,3,4]=>2
[5,2,6,1,4,3]=>1
[5,2,6,3,1,4]=>1
[5,2,6,3,4,1]=>1
[5,2,6,4,1,3]=>1
[5,2,6,4,3,1]=>1
[5,3,1,2,4,6]=>4
[5,3,1,2,6,4]=>3
[5,3,1,4,2,6]=>2
[5,3,1,4,6,2]=>2
[5,3,1,6,2,4]=>1
[5,3,1,6,4,2]=>1
[5,3,2,1,4,6]=>2
[5,3,2,1,6,4]=>1
[5,3,2,4,1,6]=>2
[5,3,2,4,6,1]=>2
[5,3,2,6,1,4]=>1
[5,3,2,6,4,1]=>1
[5,3,4,1,2,6]=>3
[5,3,4,1,6,2]=>2
[5,3,4,2,1,6]=>2
[5,3,4,2,6,1]=>2
[5,3,4,6,1,2]=>2
[5,3,4,6,2,1]=>2
[5,3,6,1,2,4]=>2
[5,3,6,1,4,2]=>1
[5,3,6,2,1,4]=>1
[5,3,6,2,4,1]=>1
[5,3,6,4,1,2]=>1
[5,3,6,4,2,1]=>1
[5,4,1,2,3,6]=>4
[5,4,1,2,6,3]=>3
[5,4,1,3,2,6]=>2
[5,4,1,3,6,2]=>2
[5,4,1,6,2,3]=>1
[5,4,1,6,3,2]=>1
[5,4,2,1,3,6]=>2
[5,4,2,1,6,3]=>1
[5,4,2,3,1,6]=>2
[5,4,2,3,6,1]=>2
[5,4,2,6,1,3]=>1
[5,4,2,6,3,1]=>1
[5,4,3,1,2,6]=>2
[5,4,3,1,6,2]=>1
[5,4,3,2,1,6]=>1
[5,4,3,2,6,1]=>1
[5,4,3,6,1,2]=>1
[5,4,3,6,2,1]=>1
[5,4,6,1,2,3]=>2
[5,4,6,1,3,2]=>1
[5,4,6,2,1,3]=>1
[5,4,6,2,3,1]=>1
[5,4,6,3,1,2]=>1
[5,4,6,3,2,1]=>1
[5,6,1,2,3,4]=>3
[5,6,1,2,4,3]=>2
[5,6,1,3,2,4]=>1
[5,6,1,3,4,2]=>1
[5,6,1,4,2,3]=>0
[5,6,1,4,3,2]=>0
[5,6,2,1,3,4]=>1
[5,6,2,1,4,3]=>0
[5,6,2,3,1,4]=>1
[5,6,2,3,4,1]=>1
[5,6,2,4,1,3]=>0
[5,6,2,4,3,1]=>0
[5,6,3,1,2,4]=>1
[5,6,3,1,4,2]=>0
[5,6,3,2,1,4]=>0
[5,6,3,2,4,1]=>0
[5,6,3,4,1,2]=>0
[5,6,3,4,2,1]=>0
[5,6,4,1,2,3]=>1
[5,6,4,1,3,2]=>0
[5,6,4,2,1,3]=>0
[5,6,4,2,3,1]=>0
[5,6,4,3,1,2]=>0
[5,6,4,3,2,1]=>0
[6,1,2,3,4,5]=>6
[6,1,2,3,5,4]=>5
[6,1,2,4,3,5]=>4
[6,1,2,4,5,3]=>4
[6,1,2,5,3,4]=>3
[6,1,2,5,4,3]=>3
[6,1,3,2,4,5]=>3
[6,1,3,2,5,4]=>2
[6,1,3,4,2,5]=>3
[6,1,3,4,5,2]=>3
[6,1,3,5,2,4]=>2
[6,1,3,5,4,2]=>2
[6,1,4,2,3,5]=>2
[6,1,4,2,5,3]=>1
[6,1,4,3,2,5]=>1
[6,1,4,3,5,2]=>1
[6,1,4,5,2,3]=>1
[6,1,4,5,3,2]=>1
[6,1,5,2,3,4]=>1
[6,1,5,2,4,3]=>0
[6,1,5,3,2,4]=>0
[6,1,5,3,4,2]=>0
[6,1,5,4,2,3]=>0
[6,1,5,4,3,2]=>0
[6,2,1,3,4,5]=>3
[6,2,1,3,5,4]=>2
[6,2,1,4,3,5]=>1
[6,2,1,4,5,3]=>1
[6,2,1,5,3,4]=>0
[6,2,1,5,4,3]=>0
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>2
[6,2,3,4,1,5]=>3
[6,2,3,4,5,1]=>3
[6,2,3,5,1,4]=>2
[6,2,3,5,4,1]=>2
[6,2,4,1,3,5]=>2
[6,2,4,1,5,3]=>1
[6,2,4,3,1,5]=>1
[6,2,4,3,5,1]=>1
[6,2,4,5,1,3]=>1
[6,2,4,5,3,1]=>1
[6,2,5,1,3,4]=>1
[6,2,5,1,4,3]=>0
[6,2,5,3,1,4]=>0
[6,2,5,3,4,1]=>0
[6,2,5,4,1,3]=>0
[6,2,5,4,3,1]=>0
[6,3,1,2,4,5]=>3
[6,3,1,2,5,4]=>2
[6,3,1,4,2,5]=>1
[6,3,1,4,5,2]=>1
[6,3,1,5,2,4]=>0
[6,3,1,5,4,2]=>0
[6,3,2,1,4,5]=>1
[6,3,2,1,5,4]=>0
[6,3,2,4,1,5]=>1
[6,3,2,4,5,1]=>1
[6,3,2,5,1,4]=>0
[6,3,2,5,4,1]=>0
[6,3,4,1,2,5]=>2
[6,3,4,1,5,2]=>1
[6,3,4,2,1,5]=>1
[6,3,4,2,5,1]=>1
[6,3,4,5,1,2]=>1
[6,3,4,5,2,1]=>1
[6,3,5,1,2,4]=>1
[6,3,5,1,4,2]=>0
[6,3,5,2,1,4]=>0
[6,3,5,2,4,1]=>0
[6,3,5,4,1,2]=>0
[6,3,5,4,2,1]=>0
[6,4,1,2,3,5]=>3
[6,4,1,2,5,3]=>2
[6,4,1,3,2,5]=>1
[6,4,1,3,5,2]=>1
[6,4,1,5,2,3]=>0
[6,4,1,5,3,2]=>0
[6,4,2,1,3,5]=>1
[6,4,2,1,5,3]=>0
[6,4,2,3,1,5]=>1
[6,4,2,3,5,1]=>1
[6,4,2,5,1,3]=>0
[6,4,2,5,3,1]=>0
[6,4,3,1,2,5]=>1
[6,4,3,1,5,2]=>0
[6,4,3,2,1,5]=>0
[6,4,3,2,5,1]=>0
[6,4,3,5,1,2]=>0
[6,4,3,5,2,1]=>0
[6,4,5,1,2,3]=>1
[6,4,5,1,3,2]=>0
[6,4,5,2,1,3]=>0
[6,4,5,2,3,1]=>0
[6,4,5,3,1,2]=>0
[6,4,5,3,2,1]=>0
[6,5,1,2,3,4]=>3
[6,5,1,2,4,3]=>2
[6,5,1,3,2,4]=>1
[6,5,1,3,4,2]=>1
[6,5,1,4,2,3]=>0
[6,5,1,4,3,2]=>0
[6,5,2,1,3,4]=>1
[6,5,2,1,4,3]=>0
[6,5,2,3,1,4]=>1
[6,5,2,3,4,1]=>1
[6,5,2,4,1,3]=>0
[6,5,2,4,3,1]=>0
[6,5,3,1,2,4]=>1
[6,5,3,1,4,2]=>0
[6,5,3,2,1,4]=>0
[6,5,3,2,4,1]=>0
[6,5,3,4,1,2]=>0
[6,5,3,4,2,1]=>0
[6,5,4,1,2,3]=>1
[6,5,4,1,3,2]=>0
[6,5,4,2,1,3]=>0
[6,5,4,2,3,1]=>0
[6,5,4,3,1,2]=>0
[6,5,4,3,2,1]=>0
[1,2,3,4,5,6,7]=>0
[1,2,3,4,5,7,6]=>0
[1,2,3,4,6,5,7]=>1
[1,2,3,4,6,7,5]=>0
[1,2,3,4,7,5,6]=>0
[1,2,3,4,7,6,5]=>0
[1,2,3,5,4,6,7]=>2
[1,2,3,5,4,7,6]=>2
[1,2,3,5,6,4,7]=>1
[1,2,3,5,6,7,4]=>0
[1,2,3,5,7,4,6]=>0
[1,2,3,5,7,6,4]=>0
[1,2,3,6,4,5,7]=>2
[1,2,3,6,4,7,5]=>1
[1,2,3,6,5,4,7]=>1
[1,2,3,6,5,7,4]=>1
[1,2,3,6,7,4,5]=>0
[1,2,3,6,7,5,4]=>0
[1,2,3,7,4,5,6]=>1
[1,2,3,7,4,6,5]=>0
[1,2,3,7,5,4,6]=>0
[1,2,3,7,5,6,4]=>0
[1,2,3,7,6,4,5]=>0
[1,2,3,7,6,5,4]=>0
[1,2,4,3,5,6,7]=>3
[1,2,4,3,5,7,6]=>3
[1,2,4,3,6,5,7]=>4
[1,2,4,3,6,7,5]=>3
[1,2,4,3,7,5,6]=>3
[1,2,4,3,7,6,5]=>3
[1,2,4,5,3,6,7]=>2
[1,2,4,5,3,7,6]=>2
[1,2,4,5,6,3,7]=>1
[1,2,4,5,6,7,3]=>0
[1,2,4,5,7,3,6]=>0
[1,2,4,5,7,6,3]=>0
[1,2,4,6,3,5,7]=>2
[1,2,4,6,3,7,5]=>1
[1,2,4,6,5,3,7]=>1
[1,2,4,6,5,7,3]=>1
[1,2,4,6,7,3,5]=>0
[1,2,4,6,7,5,3]=>0
[1,2,4,7,3,5,6]=>1
[1,2,4,7,3,6,5]=>0
[1,2,4,7,5,3,6]=>0
[1,2,4,7,5,6,3]=>0
[1,2,4,7,6,3,5]=>0
[1,2,4,7,6,5,3]=>0
[1,2,5,3,4,6,7]=>4
[1,2,5,3,4,7,6]=>4
[1,2,5,3,6,4,7]=>3
[1,2,5,3,6,7,4]=>2
[1,2,5,3,7,4,6]=>2
[1,2,5,3,7,6,4]=>2
[1,2,5,4,3,6,7]=>2
[1,2,5,4,3,7,6]=>2
[1,2,5,4,6,3,7]=>3
[1,2,5,4,6,7,3]=>2
[1,2,5,4,7,3,6]=>2
[1,2,5,4,7,6,3]=>2
[1,2,5,6,3,4,7]=>2
[1,2,5,6,3,7,4]=>1
[1,2,5,6,4,3,7]=>1
[1,2,5,6,4,7,3]=>1
[1,2,5,6,7,3,4]=>0
[1,2,5,6,7,4,3]=>0
[1,2,5,7,3,4,6]=>1
[1,2,5,7,3,6,4]=>0
[1,2,5,7,4,3,6]=>0
[1,2,5,7,4,6,3]=>0
[1,2,5,7,6,3,4]=>0
[1,2,5,7,6,4,3]=>0
[1,2,6,3,4,5,7]=>4
[1,2,6,3,4,7,5]=>3
[1,2,6,3,5,4,7]=>2
[1,2,6,3,5,7,4]=>2
[1,2,6,3,7,4,5]=>1
[1,2,6,3,7,5,4]=>1
[1,2,6,4,3,5,7]=>2
[1,2,6,4,3,7,5]=>1
[1,2,6,4,5,3,7]=>2
[1,2,6,4,5,7,3]=>2
[1,2,6,4,7,3,5]=>1
[1,2,6,4,7,5,3]=>1
[1,2,6,5,3,4,7]=>2
[1,2,6,5,3,7,4]=>1
[1,2,6,5,4,3,7]=>1
[1,2,6,5,4,7,3]=>1
[1,2,6,5,7,3,4]=>1
[1,2,6,5,7,4,3]=>1
[1,2,6,7,3,4,5]=>1
[1,2,6,7,3,5,4]=>0
[1,2,6,7,4,3,5]=>0
[1,2,6,7,4,5,3]=>0
[1,2,6,7,5,3,4]=>0
[1,2,6,7,5,4,3]=>0
[1,2,7,3,4,5,6]=>3
[1,2,7,3,4,6,5]=>2
[1,2,7,3,5,4,6]=>1
[1,2,7,3,5,6,4]=>1
[1,2,7,3,6,4,5]=>0
[1,2,7,3,6,5,4]=>0
[1,2,7,4,3,5,6]=>1
[1,2,7,4,3,6,5]=>0
[1,2,7,4,5,3,6]=>1
[1,2,7,4,5,6,3]=>1
[1,2,7,4,6,3,5]=>0
[1,2,7,4,6,5,3]=>0
[1,2,7,5,3,4,6]=>1
[1,2,7,5,3,6,4]=>0
[1,2,7,5,4,3,6]=>0
[1,2,7,5,4,6,3]=>0
[1,2,7,5,6,3,4]=>0
[1,2,7,5,6,4,3]=>0
[1,2,7,6,3,4,5]=>1
[1,2,7,6,3,5,4]=>0
[1,2,7,6,4,3,5]=>0
[1,2,7,6,4,5,3]=>0
[1,2,7,6,5,3,4]=>0
[1,2,7,6,5,4,3]=>0
[1,3,2,4,5,6,7]=>4
[1,3,2,4,5,7,6]=>4
[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]=>4
[1,3,2,5,4,6,7]=>6
[1,3,2,5,4,7,6]=>6
[1,3,2,5,6,4,7]=>5
[1,3,2,5,6,7,4]=>4
[1,3,2,5,7,4,6]=>4
[1,3,2,5,7,6,4]=>4
[1,3,2,6,4,5,7]=>6
[1,3,2,6,4,7,5]=>5
[1,3,2,6,5,4,7]=>5
[1,3,2,6,5,7,4]=>5
[1,3,2,6,7,4,5]=>4
[1,3,2,6,7,5,4]=>4
[1,3,2,7,4,5,6]=>5
[1,3,2,7,4,6,5]=>4
[1,3,2,7,5,4,6]=>4
[1,3,2,7,5,6,4]=>4
[1,3,2,7,6,4,5]=>4
[1,3,2,7,6,5,4]=>4
[1,3,4,2,5,6,7]=>3
[1,3,4,2,5,7,6]=>3
[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]=>3
[1,3,4,5,2,6,7]=>2
[1,3,4,5,2,7,6]=>2
[1,3,4,5,6,2,7]=>1
[1,3,4,5,6,7,2]=>0
[1,3,4,5,7,2,6]=>0
[1,3,4,5,7,6,2]=>0
[1,3,4,6,2,5,7]=>2
[1,3,4,6,2,7,5]=>1
[1,3,4,6,5,2,7]=>1
[1,3,4,6,5,7,2]=>1
[1,3,4,6,7,2,5]=>0
[1,3,4,6,7,5,2]=>0
[1,3,4,7,2,5,6]=>1
[1,3,4,7,2,6,5]=>0
[1,3,4,7,5,2,6]=>0
[1,3,4,7,5,6,2]=>0
[1,3,4,7,6,2,5]=>0
[1,3,4,7,6,5,2]=>0
[1,3,5,2,4,6,7]=>4
[1,3,5,2,4,7,6]=>4
[1,3,5,2,6,4,7]=>3
[1,3,5,2,6,7,4]=>2
[1,3,5,2,7,4,6]=>2
[1,3,5,2,7,6,4]=>2
[1,3,5,4,2,6,7]=>2
[1,3,5,4,2,7,6]=>2
[1,3,5,4,6,2,7]=>3
[1,3,5,4,6,7,2]=>2
[1,3,5,4,7,2,6]=>2
[1,3,5,4,7,6,2]=>2
[1,3,5,6,2,4,7]=>2
[1,3,5,6,2,7,4]=>1
[1,3,5,6,4,2,7]=>1
[1,3,5,6,4,7,2]=>1
[1,3,5,6,7,2,4]=>0
[1,3,5,6,7,4,2]=>0
[1,3,5,7,2,4,6]=>1
[1,3,5,7,2,6,4]=>0
[1,3,5,7,4,2,6]=>0
[1,3,5,7,4,6,2]=>0
[1,3,5,7,6,2,4]=>0
[1,3,5,7,6,4,2]=>0
[1,3,6,2,4,5,7]=>4
[1,3,6,2,4,7,5]=>3
[1,3,6,2,5,4,7]=>2
[1,3,6,2,5,7,4]=>2
[1,3,6,2,7,4,5]=>1
[1,3,6,2,7,5,4]=>1
[1,3,6,4,2,5,7]=>2
[1,3,6,4,2,7,5]=>1
[1,3,6,4,5,2,7]=>2
[1,3,6,4,5,7,2]=>2
[1,3,6,4,7,2,5]=>1
[1,3,6,4,7,5,2]=>1
[1,3,6,5,2,4,7]=>2
[1,3,6,5,2,7,4]=>1
[1,3,6,5,4,2,7]=>1
[1,3,6,5,4,7,2]=>1
[1,3,6,5,7,2,4]=>1
[1,3,6,5,7,4,2]=>1
[1,3,6,7,2,4,5]=>1
[1,3,6,7,2,5,4]=>0
[1,3,6,7,4,2,5]=>0
[1,3,6,7,4,5,2]=>0
[1,3,6,7,5,2,4]=>0
[1,3,6,7,5,4,2]=>0
[1,3,7,2,4,5,6]=>3
[1,3,7,2,4,6,5]=>2
[1,3,7,2,5,4,6]=>1
[1,3,7,2,5,6,4]=>1
[1,3,7,2,6,4,5]=>0
[1,3,7,2,6,5,4]=>0
[1,3,7,4,2,5,6]=>1
[1,3,7,4,2,6,5]=>0
[1,3,7,4,5,2,6]=>1
[1,3,7,4,5,6,2]=>1
[1,3,7,4,6,2,5]=>0
[1,3,7,4,6,5,2]=>0
[1,3,7,5,2,4,6]=>1
[1,3,7,5,2,6,4]=>0
[1,3,7,5,4,2,6]=>0
[1,3,7,5,4,6,2]=>0
[1,3,7,5,6,2,4]=>0
[1,3,7,5,6,4,2]=>0
[1,3,7,6,2,4,5]=>1
[1,3,7,6,2,5,4]=>0
[1,3,7,6,4,2,5]=>0
[1,3,7,6,4,5,2]=>0
[1,3,7,6,5,2,4]=>0
[1,3,7,6,5,4,2]=>0
[1,4,2,3,5,6,7]=>6
[1,4,2,3,5,7,6]=>6
[1,4,2,3,6,5,7]=>7
[1,4,2,3,6,7,5]=>6
[1,4,2,3,7,5,6]=>6
[1,4,2,3,7,6,5]=>6
[1,4,2,5,3,6,7]=>5
[1,4,2,5,3,7,6]=>5
[1,4,2,5,6,3,7]=>4
[1,4,2,5,6,7,3]=>3
[1,4,2,5,7,3,6]=>3
[1,4,2,5,7,6,3]=>3
[1,4,2,6,3,5,7]=>5
[1,4,2,6,3,7,5]=>4
[1,4,2,6,5,3,7]=>4
[1,4,2,6,5,7,3]=>4
[1,4,2,6,7,3,5]=>3
[1,4,2,6,7,5,3]=>3
[1,4,2,7,3,5,6]=>4
[1,4,2,7,3,6,5]=>3
[1,4,2,7,5,3,6]=>3
[1,4,2,7,5,6,3]=>3
[1,4,2,7,6,3,5]=>3
[1,4,2,7,6,5,3]=>3
[1,4,3,2,5,6,7]=>3
[1,4,3,2,5,7,6]=>3
[1,4,3,2,6,5,7]=>4
[1,4,3,2,6,7,5]=>3
[1,4,3,2,7,5,6]=>3
[1,4,3,2,7,6,5]=>3
[1,4,3,5,2,6,7]=>5
[1,4,3,5,2,7,6]=>5
[1,4,3,5,6,2,7]=>4
[1,4,3,5,6,7,2]=>3
[1,4,3,5,7,2,6]=>3
[1,4,3,5,7,6,2]=>3
[1,4,3,6,2,5,7]=>5
[1,4,3,6,2,7,5]=>4
[1,4,3,6,5,2,7]=>4
[1,4,3,6,5,7,2]=>4
[1,4,3,6,7,2,5]=>3
[1,4,3,6,7,5,2]=>3
[1,4,3,7,2,5,6]=>4
[1,4,3,7,2,6,5]=>3
[1,4,3,7,5,2,6]=>3
[1,4,3,7,5,6,2]=>3
[1,4,3,7,6,2,5]=>3
[1,4,3,7,6,5,2]=>3
[1,4,5,2,3,6,7]=>4
[1,4,5,2,3,7,6]=>4
[1,4,5,2,6,3,7]=>3
[1,4,5,2,6,7,3]=>2
[1,4,5,2,7,3,6]=>2
[1,4,5,2,7,6,3]=>2
[1,4,5,3,2,6,7]=>2
[1,4,5,3,2,7,6]=>2
[1,4,5,3,6,2,7]=>3
[1,4,5,3,6,7,2]=>2
[1,4,5,3,7,2,6]=>2
[1,4,5,3,7,6,2]=>2
[1,4,5,6,2,3,7]=>2
[1,4,5,6,2,7,3]=>1
[1,4,5,6,3,2,7]=>1
[1,4,5,6,3,7,2]=>1
[1,4,5,6,7,2,3]=>0
[1,4,5,6,7,3,2]=>0
[1,4,5,7,2,3,6]=>1
[1,4,5,7,2,6,3]=>0
[1,4,5,7,3,2,6]=>0
[1,4,5,7,3,6,2]=>0
[1,4,5,7,6,2,3]=>0
[1,4,5,7,6,3,2]=>0
[1,4,6,2,3,5,7]=>4
[1,4,6,2,3,7,5]=>3
[1,4,6,2,5,3,7]=>2
[1,4,6,2,5,7,3]=>2
[1,4,6,2,7,3,5]=>1
[1,4,6,2,7,5,3]=>1
[1,4,6,3,2,5,7]=>2
[1,4,6,3,2,7,5]=>1
[1,4,6,3,5,2,7]=>2
[1,4,6,3,5,7,2]=>2
[1,4,6,3,7,2,5]=>1
[1,4,6,3,7,5,2]=>1
[1,4,6,5,2,3,7]=>2
[1,4,6,5,2,7,3]=>1
[1,4,6,5,3,2,7]=>1
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
References
[1] Berman, Y., Tenner, B. E. Pattern-functions, statistics, and shallow permutations arXiv:2110.11146
Code
def vincular_occurrences_iterator(perm, pat, columns):
perm = Permutation(perm)
for pos in perm.pattern_positions(pat):
if all(pos[i-1]+1 == pos[i] for i in columns):
yield tuple(pos)
from sage.combinat.permutation import to_standard
def arrow_occurrences_iterator(perm, pat, columns, arrows):
perm = Permutation(perm)
s = set(pat + [p for p, _ in arrows] + [q for _, q in arrows])
k = max(s)
assert min(s) == 1 and len(s) == k
nu = to_standard(pat)
sigma = perm.fundamental_transformation_inverse()
for pos in vincular_occurrences_iterator(perm, nu, columns):
x = [None]*k
for i, a in enumerate(pos):
x[pat[i]-1] = perm[a]
for p, q in arrows:
if x[p-1] is None and x[q-1] is None:
raise ValueError("doesn't work for %s %s %s %s" % (perm, pat, columns, arrows))
if x[p-1] is None:
x[p-1] = sigma.inverse()(x[q-1])
elif x[q-1] is None:
x[q-1] = sigma(x[p-1])
if (all(x[i] < x[i+1] for i in range(len(x)-1))
and all(sigma(x[p-1]) == x[q-1] for p, q in arrows)):
yield tuple(pos)
def arrow_occurrences(perm, pat, columns, arrows):
return list(arrow_occurrences_iterator(perm, pat, columns, arrows))
def statistic(pi):
return len(arrow_occurrences(pi, [1,3], [], [[1,2]]))
Created
Oct 23, 2021 at 00:39 by Martin Rubey
Updated
Oct 23, 2021 at 00:39 by Martin Rubey
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!