Identifier
- St001874: 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]=>1
[3,1,2]=>1
[3,2,1]=>3
[1,2,3,4]=>0
[1,2,4,3]=>1
[1,3,2,4]=>1
[1,3,4,2]=>1
[1,4,2,3]=>1
[1,4,3,2]=>3
[2,1,3,4]=>1
[2,1,4,3]=>2
[2,3,1,4]=>1
[2,3,4,1]=>1
[2,4,1,3]=>2
[2,4,3,1]=>3
[3,1,2,4]=>1
[3,1,4,2]=>2
[3,2,1,4]=>3
[3,2,4,1]=>3
[3,4,1,2]=>2
[3,4,2,1]=>3
[4,1,2,3]=>1
[4,1,3,2]=>3
[4,2,1,3]=>3
[4,2,3,1]=>3
[4,3,1,2]=>3
[4,3,2,1]=>6
[1,2,3,4,5]=>0
[1,2,3,5,4]=>1
[1,2,4,3,5]=>1
[1,2,4,5,3]=>1
[1,2,5,3,4]=>1
[1,2,5,4,3]=>3
[1,3,2,4,5]=>1
[1,3,2,5,4]=>2
[1,3,4,2,5]=>1
[1,3,4,5,2]=>1
[1,3,5,2,4]=>2
[1,3,5,4,2]=>3
[1,4,2,3,5]=>1
[1,4,2,5,3]=>2
[1,4,3,2,5]=>3
[1,4,3,5,2]=>3
[1,4,5,2,3]=>2
[1,4,5,3,2]=>3
[1,5,2,3,4]=>1
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>3
[1,5,4,2,3]=>3
[1,5,4,3,2]=>6
[2,1,3,4,5]=>1
[2,1,3,5,4]=>2
[2,1,4,3,5]=>2
[2,1,4,5,3]=>2
[2,1,5,3,4]=>2
[2,1,5,4,3]=>4
[2,3,1,4,5]=>1
[2,3,1,5,4]=>2
[2,3,4,1,5]=>1
[2,3,4,5,1]=>1
[2,3,5,1,4]=>2
[2,3,5,4,1]=>3
[2,4,1,3,5]=>2
[2,4,1,5,3]=>2
[2,4,3,1,5]=>3
[2,4,3,5,1]=>3
[2,4,5,1,3]=>2
[2,4,5,3,1]=>3
[2,5,1,3,4]=>2
[2,5,1,4,3]=>4
[2,5,3,1,4]=>3
[2,5,3,4,1]=>3
[2,5,4,1,3]=>4
[2,5,4,3,1]=>6
[3,1,2,4,5]=>1
[3,1,2,5,4]=>2
[3,1,4,2,5]=>2
[3,1,4,5,2]=>2
[3,1,5,2,4]=>2
[3,1,5,4,2]=>4
[3,2,1,4,5]=>3
[3,2,1,5,4]=>4
[3,2,4,1,5]=>3
[3,2,4,5,1]=>3
[3,2,5,1,4]=>4
[3,2,5,4,1]=>4
[3,4,1,2,5]=>2
[3,4,1,5,2]=>2
[3,4,2,1,5]=>3
[3,4,2,5,1]=>3
[3,4,5,1,2]=>2
[3,4,5,2,1]=>3
[3,5,1,2,4]=>2
[3,5,1,4,2]=>4
[3,5,2,1,4]=>4
[3,5,2,4,1]=>4
[3,5,4,1,2]=>4
[3,5,4,2,1]=>6
[4,1,2,3,5]=>1
[4,1,2,5,3]=>2
[4,1,3,2,5]=>3
[4,1,3,5,2]=>3
[4,1,5,2,3]=>2
[4,1,5,3,2]=>4
[4,2,1,3,5]=>3
[4,2,1,5,3]=>4
[4,2,3,1,5]=>3
[4,2,3,5,1]=>3
[4,2,5,1,3]=>4
[4,2,5,3,1]=>4
[4,3,1,2,5]=>3
[4,3,1,5,2]=>4
[4,3,2,1,5]=>6
[4,3,2,5,1]=>6
[4,3,5,1,2]=>4
[4,3,5,2,1]=>6
[4,5,1,2,3]=>2
[4,5,1,3,2]=>4
[4,5,2,1,3]=>4
[4,5,2,3,1]=>4
[4,5,3,1,2]=>4
[4,5,3,2,1]=>6
[5,1,2,3,4]=>1
[5,1,2,4,3]=>3
[5,1,3,2,4]=>3
[5,1,3,4,2]=>3
[5,1,4,2,3]=>3
[5,1,4,3,2]=>6
[5,2,1,3,4]=>3
[5,2,1,4,3]=>4
[5,2,3,1,4]=>3
[5,2,3,4,1]=>3
[5,2,4,1,3]=>4
[5,2,4,3,1]=>6
[5,3,1,2,4]=>3
[5,3,1,4,2]=>4
[5,3,2,1,4]=>6
[5,3,2,4,1]=>6
[5,3,4,1,2]=>4
[5,3,4,2,1]=>6
[5,4,1,2,3]=>3
[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]=>10
[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]=>1
[1,2,3,6,4,5]=>1
[1,2,3,6,5,4]=>3
[1,2,4,3,5,6]=>1
[1,2,4,3,6,5]=>2
[1,2,4,5,3,6]=>1
[1,2,4,5,6,3]=>1
[1,2,4,6,3,5]=>2
[1,2,4,6,5,3]=>3
[1,2,5,3,4,6]=>1
[1,2,5,3,6,4]=>2
[1,2,5,4,3,6]=>3
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>2
[1,2,5,6,4,3]=>3
[1,2,6,3,4,5]=>1
[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]=>3
[1,2,6,5,4,3]=>6
[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]=>2
[1,3,2,6,4,5]=>2
[1,3,2,6,5,4]=>4
[1,3,4,2,5,6]=>1
[1,3,4,2,6,5]=>2
[1,3,4,5,2,6]=>1
[1,3,4,5,6,2]=>1
[1,3,4,6,2,5]=>2
[1,3,4,6,5,2]=>3
[1,3,5,2,4,6]=>2
[1,3,5,2,6,4]=>2
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>3
[1,3,5,6,2,4]=>2
[1,3,5,6,4,2]=>3
[1,3,6,2,4,5]=>2
[1,3,6,2,5,4]=>4
[1,3,6,4,2,5]=>3
[1,3,6,4,5,2]=>3
[1,3,6,5,2,4]=>4
[1,3,6,5,4,2]=>6
[1,4,2,3,5,6]=>1
[1,4,2,3,6,5]=>2
[1,4,2,5,3,6]=>2
[1,4,2,5,6,3]=>2
[1,4,2,6,3,5]=>2
[1,4,2,6,5,3]=>4
[1,4,3,2,5,6]=>3
[1,4,3,2,6,5]=>4
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>3
[1,4,3,6,2,5]=>4
[1,4,3,6,5,2]=>4
[1,4,5,2,3,6]=>2
[1,4,5,2,6,3]=>2
[1,4,5,3,2,6]=>3
[1,4,5,3,6,2]=>3
[1,4,5,6,2,3]=>2
[1,4,5,6,3,2]=>3
[1,4,6,2,3,5]=>2
[1,4,6,2,5,3]=>4
[1,4,6,3,2,5]=>4
[1,4,6,3,5,2]=>4
[1,4,6,5,2,3]=>4
[1,4,6,5,3,2]=>6
[1,5,2,3,4,6]=>1
[1,5,2,3,6,4]=>2
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>3
[1,5,2,6,3,4]=>2
[1,5,2,6,4,3]=>4
[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]=>3
[1,5,3,6,2,4]=>4
[1,5,3,6,4,2]=>4
[1,5,4,2,3,6]=>3
[1,5,4,2,6,3]=>4
[1,5,4,3,2,6]=>6
[1,5,4,3,6,2]=>6
[1,5,4,6,2,3]=>4
[1,5,4,6,3,2]=>6
[1,5,6,2,3,4]=>2
[1,5,6,2,4,3]=>4
[1,5,6,3,2,4]=>4
[1,5,6,3,4,2]=>4
[1,5,6,4,2,3]=>4
[1,5,6,4,3,2]=>6
[1,6,2,3,4,5]=>1
[1,6,2,3,5,4]=>3
[1,6,2,4,3,5]=>3
[1,6,2,4,5,3]=>3
[1,6,2,5,3,4]=>3
[1,6,2,5,4,3]=>6
[1,6,3,2,4,5]=>3
[1,6,3,2,5,4]=>4
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>3
[1,6,3,5,2,4]=>4
[1,6,3,5,4,2]=>6
[1,6,4,2,3,5]=>3
[1,6,4,2,5,3]=>4
[1,6,4,3,2,5]=>6
[1,6,4,3,5,2]=>6
[1,6,4,5,2,3]=>4
[1,6,4,5,3,2]=>6
[1,6,5,2,3,4]=>3
[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]=>10
[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]=>2
[2,1,3,6,4,5]=>2
[2,1,3,6,5,4]=>4
[2,1,4,3,5,6]=>2
[2,1,4,3,6,5]=>3
[2,1,4,5,3,6]=>2
[2,1,4,5,6,3]=>2
[2,1,4,6,3,5]=>3
[2,1,4,6,5,3]=>4
[2,1,5,3,4,6]=>2
[2,1,5,3,6,4]=>3
[2,1,5,4,3,6]=>4
[2,1,5,4,6,3]=>4
[2,1,5,6,3,4]=>3
[2,1,5,6,4,3]=>4
[2,1,6,3,4,5]=>2
[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]=>7
[2,3,1,4,5,6]=>1
[2,3,1,4,6,5]=>2
[2,3,1,5,4,6]=>2
[2,3,1,5,6,4]=>2
[2,3,1,6,4,5]=>2
[2,3,1,6,5,4]=>4
[2,3,4,1,5,6]=>1
[2,3,4,1,6,5]=>2
[2,3,4,5,1,6]=>1
[2,3,4,5,6,1]=>1
[2,3,4,6,1,5]=>2
[2,3,4,6,5,1]=>3
[2,3,5,1,4,6]=>2
[2,3,5,1,6,4]=>2
[2,3,5,4,1,6]=>3
[2,3,5,4,6,1]=>3
[2,3,5,6,1,4]=>2
[2,3,5,6,4,1]=>3
[2,3,6,1,4,5]=>2
[2,3,6,1,5,4]=>4
[2,3,6,4,1,5]=>3
[2,3,6,4,5,1]=>3
[2,3,6,5,1,4]=>4
[2,3,6,5,4,1]=>6
[2,4,1,3,5,6]=>2
[2,4,1,3,6,5]=>3
[2,4,1,5,3,6]=>2
[2,4,1,5,6,3]=>2
[2,4,1,6,3,5]=>3
[2,4,1,6,5,3]=>4
[2,4,3,1,5,6]=>3
[2,4,3,1,6,5]=>4
[2,4,3,5,1,6]=>3
[2,4,3,5,6,1]=>3
[2,4,3,6,1,5]=>4
[2,4,3,6,5,1]=>4
[2,4,5,1,3,6]=>2
[2,4,5,1,6,3]=>2
[2,4,5,3,1,6]=>3
[2,4,5,3,6,1]=>3
[2,4,5,6,1,3]=>2
[2,4,5,6,3,1]=>3
[2,4,6,1,3,5]=>3
[2,4,6,1,5,3]=>4
[2,4,6,3,1,5]=>4
[2,4,6,3,5,1]=>4
[2,4,6,5,1,3]=>4
[2,4,6,5,3,1]=>6
[2,5,1,3,4,6]=>2
[2,5,1,3,6,4]=>3
[2,5,1,4,3,6]=>4
[2,5,1,4,6,3]=>4
[2,5,1,6,3,4]=>3
[2,5,1,6,4,3]=>4
[2,5,3,1,4,6]=>3
[2,5,3,1,6,4]=>4
[2,5,3,4,1,6]=>3
[2,5,3,4,6,1]=>3
[2,5,3,6,1,4]=>4
[2,5,3,6,4,1]=>4
[2,5,4,1,3,6]=>4
[2,5,4,1,6,3]=>4
[2,5,4,3,1,6]=>6
[2,5,4,3,6,1]=>6
[2,5,4,6,1,3]=>4
[2,5,4,6,3,1]=>6
[2,5,6,1,3,4]=>3
[2,5,6,1,4,3]=>4
[2,5,6,3,1,4]=>4
[2,5,6,3,4,1]=>4
[2,5,6,4,1,3]=>4
[2,5,6,4,3,1]=>6
[2,6,1,3,4,5]=>2
[2,6,1,3,5,4]=>4
[2,6,1,4,3,5]=>4
[2,6,1,4,5,3]=>4
[2,6,1,5,3,4]=>4
[2,6,1,5,4,3]=>7
[2,6,3,1,4,5]=>3
[2,6,3,1,5,4]=>4
[2,6,3,4,1,5]=>3
[2,6,3,4,5,1]=>3
[2,6,3,5,1,4]=>4
[2,6,3,5,4,1]=>6
[2,6,4,1,3,5]=>4
[2,6,4,1,5,3]=>4
[2,6,4,3,1,5]=>6
[2,6,4,3,5,1]=>6
[2,6,4,5,1,3]=>4
[2,6,4,5,3,1]=>6
[2,6,5,1,3,4]=>4
[2,6,5,1,4,3]=>7
[2,6,5,3,1,4]=>6
[2,6,5,3,4,1]=>6
[2,6,5,4,1,3]=>7
[2,6,5,4,3,1]=>10
[3,1,2,4,5,6]=>1
[3,1,2,4,6,5]=>2
[3,1,2,5,4,6]=>2
[3,1,2,5,6,4]=>2
[3,1,2,6,4,5]=>2
[3,1,2,6,5,4]=>4
[3,1,4,2,5,6]=>2
[3,1,4,2,6,5]=>3
[3,1,4,5,2,6]=>2
[3,1,4,5,6,2]=>2
[3,1,4,6,2,5]=>3
[3,1,4,6,5,2]=>4
[3,1,5,2,4,6]=>2
[3,1,5,2,6,4]=>3
[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]=>4
[3,1,6,2,4,5]=>2
[3,1,6,2,5,4]=>4
[3,1,6,4,2,5]=>4
[3,1,6,4,5,2]=>4
[3,1,6,5,2,4]=>4
[3,1,6,5,4,2]=>7
[3,2,1,4,5,6]=>3
[3,2,1,4,6,5]=>4
[3,2,1,5,4,6]=>4
[3,2,1,5,6,4]=>4
[3,2,1,6,4,5]=>4
[3,2,1,6,5,4]=>6
[3,2,4,1,5,6]=>3
[3,2,4,1,6,5]=>4
[3,2,4,5,1,6]=>3
[3,2,4,5,6,1]=>3
[3,2,4,6,1,5]=>4
[3,2,4,6,5,1]=>4
[3,2,5,1,4,6]=>4
[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]=>4
[3,2,5,6,4,1]=>4
[3,2,6,1,4,5]=>4
[3,2,6,1,5,4]=>6
[3,2,6,4,1,5]=>4
[3,2,6,4,5,1]=>4
[3,2,6,5,1,4]=>6
[3,2,6,5,4,1]=>7
[3,4,1,2,5,6]=>2
[3,4,1,2,6,5]=>3
[3,4,1,5,2,6]=>2
[3,4,1,5,6,2]=>2
[3,4,1,6,2,5]=>3
[3,4,1,6,5,2]=>4
[3,4,2,1,5,6]=>3
[3,4,2,1,6,5]=>4
[3,4,2,5,1,6]=>3
[3,4,2,5,6,1]=>3
[3,4,2,6,1,5]=>4
[3,4,2,6,5,1]=>4
[3,4,5,1,2,6]=>2
[3,4,5,1,6,2]=>2
[3,4,5,2,1,6]=>3
[3,4,5,2,6,1]=>3
[3,4,5,6,1,2]=>2
[3,4,5,6,2,1]=>3
[3,4,6,1,2,5]=>3
[3,4,6,1,5,2]=>4
[3,4,6,2,1,5]=>4
[3,4,6,2,5,1]=>4
[3,4,6,5,1,2]=>4
[3,4,6,5,2,1]=>6
[3,5,1,2,4,6]=>2
[3,5,1,2,6,4]=>3
[3,5,1,4,2,6]=>4
[3,5,1,4,6,2]=>4
[3,5,1,6,2,4]=>3
[3,5,1,6,4,2]=>4
[3,5,2,1,4,6]=>4
[3,5,2,1,6,4]=>4
[3,5,2,4,1,6]=>4
[3,5,2,4,6,1]=>4
[3,5,2,6,1,4]=>4
[3,5,2,6,4,1]=>4
[3,5,4,1,2,6]=>4
[3,5,4,1,6,2]=>4
[3,5,4,2,1,6]=>6
[3,5,4,2,6,1]=>6
[3,5,4,6,1,2]=>4
[3,5,4,6,2,1]=>6
[3,5,6,1,2,4]=>3
[3,5,6,1,4,2]=>4
[3,5,6,2,1,4]=>4
[3,5,6,2,4,1]=>4
[3,5,6,4,1,2]=>4
[3,5,6,4,2,1]=>6
[3,6,1,2,4,5]=>2
[3,6,1,2,5,4]=>4
[3,6,1,4,2,5]=>4
[3,6,1,4,5,2]=>4
[3,6,1,5,2,4]=>4
[3,6,1,5,4,2]=>7
[3,6,2,1,4,5]=>4
[3,6,2,1,5,4]=>6
[3,6,2,4,1,5]=>4
[3,6,2,4,5,1]=>4
[3,6,2,5,1,4]=>6
[3,6,2,5,4,1]=>7
[3,6,4,1,2,5]=>4
[3,6,4,1,5,2]=>4
[3,6,4,2,1,5]=>6
[3,6,4,2,5,1]=>6
[3,6,4,5,1,2]=>4
[3,6,4,5,2,1]=>6
[3,6,5,1,2,4]=>4
[3,6,5,1,4,2]=>7
[3,6,5,2,1,4]=>7
[3,6,5,2,4,1]=>7
[3,6,5,4,1,2]=>7
[3,6,5,4,2,1]=>10
[4,1,2,3,5,6]=>1
[4,1,2,3,6,5]=>2
[4,1,2,5,3,6]=>2
[4,1,2,5,6,3]=>2
[4,1,2,6,3,5]=>2
[4,1,2,6,5,3]=>4
[4,1,3,2,5,6]=>3
[4,1,3,2,6,5]=>4
[4,1,3,5,2,6]=>3
[4,1,3,5,6,2]=>3
[4,1,3,6,2,5]=>4
[4,1,3,6,5,2]=>4
[4,1,5,2,3,6]=>2
[4,1,5,2,6,3]=>3
[4,1,5,3,2,6]=>4
[4,1,5,3,6,2]=>4
[4,1,5,6,2,3]=>3
[4,1,5,6,3,2]=>4
[4,1,6,2,3,5]=>2
[4,1,6,2,5,3]=>4
[4,1,6,3,2,5]=>4
[4,1,6,3,5,2]=>4
[4,1,6,5,2,3]=>4
[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]=>4
[4,2,1,6,3,5]=>4
[4,2,1,6,5,3]=>6
[4,2,3,1,5,6]=>3
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>3
[4,2,3,5,6,1]=>3
[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]=>4
[4,2,5,3,1,6]=>4
[4,2,5,3,6,1]=>4
[4,2,5,6,1,3]=>4
[4,2,5,6,3,1]=>4
[4,2,6,1,3,5]=>4
[4,2,6,1,5,3]=>6
[4,2,6,3,1,5]=>4
[4,2,6,3,5,1]=>4
[4,2,6,5,1,3]=>6
[4,2,6,5,3,1]=>7
[4,3,1,2,5,6]=>3
[4,3,1,2,6,5]=>4
[4,3,1,5,2,6]=>4
[4,3,1,5,6,2]=>4
[4,3,1,6,2,5]=>4
[4,3,1,6,5,2]=>6
[4,3,2,1,5,6]=>6
[4,3,2,1,6,5]=>7
[4,3,2,5,1,6]=>6
[4,3,2,5,6,1]=>6
[4,3,2,6,1,5]=>7
[4,3,2,6,5,1]=>7
[4,3,5,1,2,6]=>4
[4,3,5,1,6,2]=>4
[4,3,5,2,1,6]=>6
[4,3,5,2,6,1]=>6
[4,3,5,6,1,2]=>4
[4,3,5,6,2,1]=>6
[4,3,6,1,2,5]=>4
[4,3,6,1,5,2]=>6
[4,3,6,2,1,5]=>7
[4,3,6,2,5,1]=>7
[4,3,6,5,1,2]=>6
[4,3,6,5,2,1]=>7
[4,5,1,2,3,6]=>2
[4,5,1,2,6,3]=>3
[4,5,1,3,2,6]=>4
[4,5,1,3,6,2]=>4
[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]=>4
[4,5,2,3,1,6]=>4
[4,5,2,3,6,1]=>4
[4,5,2,6,1,3]=>4
[4,5,2,6,3,1]=>4
[4,5,3,1,2,6]=>4
[4,5,3,1,6,2]=>4
[4,5,3,2,1,6]=>6
[4,5,3,2,6,1]=>6
[4,5,3,6,1,2]=>4
[4,5,3,6,2,1]=>6
[4,5,6,1,2,3]=>3
[4,5,6,1,3,2]=>4
[4,5,6,2,1,3]=>4
[4,5,6,2,3,1]=>4
[4,5,6,3,1,2]=>4
[4,5,6,3,2,1]=>6
[4,6,1,2,3,5]=>2
[4,6,1,2,5,3]=>4
[4,6,1,3,2,5]=>4
[4,6,1,3,5,2]=>4
[4,6,1,5,2,3]=>4
[4,6,1,5,3,2]=>7
[4,6,2,1,3,5]=>4
[4,6,2,1,5,3]=>6
[4,6,2,3,1,5]=>4
[4,6,2,3,5,1]=>4
[4,6,2,5,1,3]=>6
[4,6,2,5,3,1]=>7
[4,6,3,1,2,5]=>4
[4,6,3,1,5,2]=>6
[4,6,3,2,1,5]=>7
[4,6,3,2,5,1]=>7
[4,6,3,5,1,2]=>6
[4,6,3,5,2,1]=>7
[4,6,5,1,2,3]=>4
[4,6,5,1,3,2]=>7
[4,6,5,2,1,3]=>7
[4,6,5,2,3,1]=>7
[4,6,5,3,1,2]=>7
[4,6,5,3,2,1]=>10
[5,1,2,3,4,6]=>1
[5,1,2,3,6,4]=>2
[5,1,2,4,3,6]=>3
[5,1,2,4,6,3]=>3
[5,1,2,6,3,4]=>2
[5,1,2,6,4,3]=>4
[5,1,3,2,4,6]=>3
[5,1,3,2,6,4]=>4
[5,1,3,4,2,6]=>3
[5,1,3,4,6,2]=>3
[5,1,3,6,2,4]=>4
[5,1,3,6,4,2]=>4
[5,1,4,2,3,6]=>3
[5,1,4,2,6,3]=>4
[5,1,4,3,2,6]=>6
[5,1,4,3,6,2]=>6
[5,1,4,6,2,3]=>4
[5,1,4,6,3,2]=>6
[5,1,6,2,3,4]=>2
[5,1,6,2,4,3]=>4
[5,1,6,3,2,4]=>4
[5,1,6,3,4,2]=>4
[5,1,6,4,2,3]=>4
[5,1,6,4,3,2]=>7
[5,2,1,3,4,6]=>3
[5,2,1,3,6,4]=>4
[5,2,1,4,3,6]=>4
[5,2,1,4,6,3]=>4
[5,2,1,6,3,4]=>4
[5,2,1,6,4,3]=>6
[5,2,3,1,4,6]=>3
[5,2,3,1,6,4]=>4
[5,2,3,4,1,6]=>3
[5,2,3,4,6,1]=>3
[5,2,3,6,1,4]=>4
[5,2,3,6,4,1]=>4
[5,2,4,1,3,6]=>4
[5,2,4,1,6,3]=>4
[5,2,4,3,1,6]=>6
[5,2,4,3,6,1]=>6
[5,2,4,6,1,3]=>4
[5,2,4,6,3,1]=>6
[5,2,6,1,3,4]=>4
[5,2,6,1,4,3]=>6
[5,2,6,3,1,4]=>4
[5,2,6,3,4,1]=>4
[5,2,6,4,1,3]=>6
[5,2,6,4,3,1]=>7
[5,3,1,2,4,6]=>3
[5,3,1,2,6,4]=>4
[5,3,1,4,2,6]=>4
[5,3,1,4,6,2]=>4
[5,3,1,6,2,4]=>4
[5,3,1,6,4,2]=>6
[5,3,2,1,4,6]=>6
[5,3,2,1,6,4]=>7
[5,3,2,4,1,6]=>6
[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]=>4
[5,3,4,1,6,2]=>4
[5,3,4,2,1,6]=>6
[5,3,4,2,6,1]=>6
[5,3,4,6,1,2]=>4
[5,3,4,6,2,1]=>6
[5,3,6,1,2,4]=>4
[5,3,6,1,4,2]=>6
[5,3,6,2,1,4]=>7
[5,3,6,2,4,1]=>7
[5,3,6,4,1,2]=>6
[5,3,6,4,2,1]=>7
[5,4,1,2,3,6]=>3
[5,4,1,2,6,3]=>4
[5,4,1,3,2,6]=>6
[5,4,1,3,6,2]=>6
[5,4,1,6,2,3]=>4
[5,4,1,6,3,2]=>7
[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]=>6
[5,4,2,6,1,3]=>7
[5,4,2,6,3,1]=>7
[5,4,3,1,2,6]=>6
[5,4,3,1,6,2]=>7
[5,4,3,2,1,6]=>10
[5,4,3,2,6,1]=>10
[5,4,3,6,1,2]=>7
[5,4,3,6,2,1]=>10
[5,4,6,1,2,3]=>4
[5,4,6,1,3,2]=>7
[5,4,6,2,1,3]=>7
[5,4,6,2,3,1]=>7
[5,4,6,3,1,2]=>7
[5,4,6,3,2,1]=>10
[5,6,1,2,3,4]=>2
[5,6,1,2,4,3]=>4
[5,6,1,3,2,4]=>4
[5,6,1,3,4,2]=>4
[5,6,1,4,2,3]=>4
[5,6,1,4,3,2]=>7
[5,6,2,1,3,4]=>4
[5,6,2,1,4,3]=>6
[5,6,2,3,1,4]=>4
[5,6,2,3,4,1]=>4
[5,6,2,4,1,3]=>6
[5,6,2,4,3,1]=>7
[5,6,3,1,2,4]=>4
[5,6,3,1,4,2]=>6
[5,6,3,2,1,4]=>7
[5,6,3,2,4,1]=>7
[5,6,3,4,1,2]=>6
[5,6,3,4,2,1]=>7
[5,6,4,1,2,3]=>4
[5,6,4,1,3,2]=>7
[5,6,4,2,1,3]=>7
[5,6,4,2,3,1]=>7
[5,6,4,3,1,2]=>7
[5,6,4,3,2,1]=>10
[6,1,2,3,4,5]=>1
[6,1,2,3,5,4]=>3
[6,1,2,4,3,5]=>3
[6,1,2,4,5,3]=>3
[6,1,2,5,3,4]=>3
[6,1,2,5,4,3]=>6
[6,1,3,2,4,5]=>3
[6,1,3,2,5,4]=>4
[6,1,3,4,2,5]=>3
[6,1,3,4,5,2]=>3
[6,1,3,5,2,4]=>4
[6,1,3,5,4,2]=>6
[6,1,4,2,3,5]=>3
[6,1,4,2,5,3]=>4
[6,1,4,3,2,5]=>6
[6,1,4,3,5,2]=>6
[6,1,4,5,2,3]=>4
[6,1,4,5,3,2]=>6
[6,1,5,2,3,4]=>3
[6,1,5,2,4,3]=>6
[6,1,5,3,2,4]=>6
[6,1,5,3,4,2]=>6
[6,1,5,4,2,3]=>6
[6,1,5,4,3,2]=>10
[6,2,1,3,4,5]=>3
[6,2,1,3,5,4]=>4
[6,2,1,4,3,5]=>4
[6,2,1,4,5,3]=>4
[6,2,1,5,3,4]=>4
[6,2,1,5,4,3]=>7
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>4
[6,2,3,4,1,5]=>3
[6,2,3,4,5,1]=>3
[6,2,3,5,1,4]=>4
[6,2,3,5,4,1]=>6
[6,2,4,1,3,5]=>4
[6,2,4,1,5,3]=>4
[6,2,4,3,1,5]=>6
[6,2,4,3,5,1]=>6
[6,2,4,5,1,3]=>4
[6,2,4,5,3,1]=>6
[6,2,5,1,3,4]=>4
[6,2,5,1,4,3]=>7
[6,2,5,3,1,4]=>6
[6,2,5,3,4,1]=>6
[6,2,5,4,1,3]=>7
[6,2,5,4,3,1]=>10
[6,3,1,2,4,5]=>3
[6,3,1,2,5,4]=>4
[6,3,1,4,2,5]=>4
[6,3,1,4,5,2]=>4
[6,3,1,5,2,4]=>4
[6,3,1,5,4,2]=>7
[6,3,2,1,4,5]=>6
[6,3,2,1,5,4]=>7
[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]=>4
[6,3,4,1,5,2]=>4
[6,3,4,2,1,5]=>6
[6,3,4,2,5,1]=>6
[6,3,4,5,1,2]=>4
[6,3,4,5,2,1]=>6
[6,3,5,1,2,4]=>4
[6,3,5,1,4,2]=>7
[6,3,5,2,1,4]=>7
[6,3,5,2,4,1]=>7
[6,3,5,4,1,2]=>7
[6,3,5,4,2,1]=>10
[6,4,1,2,3,5]=>3
[6,4,1,2,5,3]=>4
[6,4,1,3,2,5]=>6
[6,4,1,3,5,2]=>6
[6,4,1,5,2,3]=>4
[6,4,1,5,3,2]=>7
[6,4,2,1,3,5]=>6
[6,4,2,1,5,3]=>7
[6,4,2,3,1,5]=>6
[6,4,2,3,5,1]=>6
[6,4,2,5,1,3]=>7
[6,4,2,5,3,1]=>7
[6,4,3,1,2,5]=>6
[6,4,3,1,5,2]=>7
[6,4,3,2,1,5]=>10
[6,4,3,2,5,1]=>10
[6,4,3,5,1,2]=>7
[6,4,3,5,2,1]=>10
[6,4,5,1,2,3]=>4
[6,4,5,1,3,2]=>7
[6,4,5,2,1,3]=>7
[6,4,5,2,3,1]=>7
[6,4,5,3,1,2]=>7
[6,4,5,3,2,1]=>10
[6,5,1,2,3,4]=>3
[6,5,1,2,4,3]=>6
[6,5,1,3,2,4]=>6
[6,5,1,3,4,2]=>6
[6,5,1,4,2,3]=>6
[6,5,1,4,3,2]=>10
[6,5,2,1,3,4]=>6
[6,5,2,1,4,3]=>7
[6,5,2,3,1,4]=>6
[6,5,2,3,4,1]=>6
[6,5,2,4,1,3]=>7
[6,5,2,4,3,1]=>10
[6,5,3,1,2,4]=>6
[6,5,3,1,4,2]=>7
[6,5,3,2,1,4]=>10
[6,5,3,2,4,1]=>10
[6,5,3,4,1,2]=>7
[6,5,3,4,2,1]=>10
[6,5,4,1,2,3]=>6
[6,5,4,1,3,2]=>10
[6,5,4,2,1,3]=>10
[6,5,4,2,3,1]=>10
[6,5,4,3,1,2]=>10
[6,5,4,3,2,1]=>15
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Description
Lusztig's a-function for the symmetric group.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.
References
[1] Björner, A., Brenti, F. Combinatorics of Coxeter groups. zbMATH:1110.05001
Code
def statistic(x):
Q = x.robinson_schensted()[1]
return sum(binomial(c, 2) for c in Q.conjugate().shape())
Created
Aug 14, 2020 at 20:27 by Rene Marczinzik
Updated
Apr 26, 2023 at 14:16 by Nadia Lafreniere
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