Identifier
- St000156: Permutations ⟶ ℤ
Values
=>
[1]=>0
[1,2]=>0
[2,1]=>1
[1,2,3]=>0
[1,3,2]=>2
[2,1,3]=>1
[2,3,1]=>3
[3,1,2]=>1
[3,2,1]=>2
[1,2,3,4]=>0
[1,2,4,3]=>3
[1,3,2,4]=>2
[1,3,4,2]=>5
[1,4,2,3]=>2
[1,4,3,2]=>3
[2,1,3,4]=>1
[2,1,4,3]=>4
[2,3,1,4]=>3
[2,3,4,1]=>6
[2,4,1,3]=>3
[2,4,3,1]=>4
[3,1,2,4]=>1
[3,1,4,2]=>4
[3,2,1,4]=>2
[3,2,4,1]=>5
[3,4,1,2]=>3
[3,4,2,1]=>4
[4,1,2,3]=>1
[4,1,3,2]=>2
[4,2,1,3]=>2
[4,2,3,1]=>3
[4,3,1,2]=>4
[4,3,2,1]=>5
[1,2,3,4,5]=>0
[1,2,3,5,4]=>4
[1,2,4,3,5]=>3
[1,2,4,5,3]=>7
[1,2,5,3,4]=>3
[1,2,5,4,3]=>4
[1,3,2,4,5]=>2
[1,3,2,5,4]=>6
[1,3,4,2,5]=>5
[1,3,4,5,2]=>9
[1,3,5,2,4]=>5
[1,3,5,4,2]=>6
[1,4,2,3,5]=>2
[1,4,2,5,3]=>6
[1,4,3,2,5]=>3
[1,4,3,5,2]=>7
[1,4,5,2,3]=>5
[1,4,5,3,2]=>6
[1,5,2,3,4]=>2
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>4
[1,5,4,2,3]=>6
[1,5,4,3,2]=>7
[2,1,3,4,5]=>1
[2,1,3,5,4]=>5
[2,1,4,3,5]=>4
[2,1,4,5,3]=>8
[2,1,5,3,4]=>4
[2,1,5,4,3]=>5
[2,3,1,4,5]=>3
[2,3,1,5,4]=>7
[2,3,4,1,5]=>6
[2,3,4,5,1]=>10
[2,3,5,1,4]=>6
[2,3,5,4,1]=>7
[2,4,1,3,5]=>3
[2,4,1,5,3]=>7
[2,4,3,1,5]=>4
[2,4,3,5,1]=>8
[2,4,5,1,3]=>6
[2,4,5,3,1]=>7
[2,5,1,3,4]=>3
[2,5,1,4,3]=>4
[2,5,3,1,4]=>4
[2,5,3,4,1]=>5
[2,5,4,1,3]=>7
[2,5,4,3,1]=>8
[3,1,2,4,5]=>1
[3,1,2,5,4]=>5
[3,1,4,2,5]=>4
[3,1,4,5,2]=>8
[3,1,5,2,4]=>4
[3,1,5,4,2]=>5
[3,2,1,4,5]=>2
[3,2,1,5,4]=>6
[3,2,4,1,5]=>5
[3,2,4,5,1]=>9
[3,2,5,1,4]=>5
[3,2,5,4,1]=>6
[3,4,1,2,5]=>3
[3,4,1,5,2]=>7
[3,4,2,1,5]=>4
[3,4,2,5,1]=>8
[3,4,5,1,2]=>6
[3,4,5,2,1]=>7
[3,5,1,2,4]=>3
[3,5,1,4,2]=>4
[3,5,2,1,4]=>4
[3,5,2,4,1]=>5
[3,5,4,1,2]=>7
[3,5,4,2,1]=>8
[4,1,2,3,5]=>1
[4,1,2,5,3]=>5
[4,1,3,2,5]=>2
[4,1,3,5,2]=>6
[4,1,5,2,3]=>4
[4,1,5,3,2]=>5
[4,2,1,3,5]=>2
[4,2,1,5,3]=>6
[4,2,3,1,5]=>3
[4,2,3,5,1]=>7
[4,2,5,1,3]=>5
[4,2,5,3,1]=>6
[4,3,1,2,5]=>4
[4,3,1,5,2]=>8
[4,3,2,1,5]=>5
[4,3,2,5,1]=>9
[4,3,5,1,2]=>7
[4,3,5,2,1]=>8
[4,5,1,2,3]=>3
[4,5,1,3,2]=>4
[4,5,2,1,3]=>4
[4,5,2,3,1]=>5
[4,5,3,1,2]=>5
[4,5,3,2,1]=>6
[5,1,2,3,4]=>1
[5,1,2,4,3]=>2
[5,1,3,2,4]=>2
[5,1,3,4,2]=>3
[5,1,4,2,3]=>5
[5,1,4,3,2]=>6
[5,2,1,3,4]=>2
[5,2,1,4,3]=>3
[5,2,3,1,4]=>3
[5,2,3,4,1]=>4
[5,2,4,1,3]=>6
[5,2,4,3,1]=>7
[5,3,1,2,4]=>4
[5,3,1,4,2]=>5
[5,3,2,1,4]=>5
[5,3,2,4,1]=>6
[5,3,4,1,2]=>8
[5,3,4,2,1]=>9
[5,4,1,2,3]=>4
[5,4,1,3,2]=>5
[5,4,2,1,3]=>5
[5,4,2,3,1]=>6
[5,4,3,1,2]=>6
[5,4,3,2,1]=>7
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>5
[1,2,3,5,4,6]=>4
[1,2,3,5,6,4]=>9
[1,2,3,6,4,5]=>4
[1,2,3,6,5,4]=>5
[1,2,4,3,5,6]=>3
[1,2,4,3,6,5]=>8
[1,2,4,5,3,6]=>7
[1,2,4,5,6,3]=>12
[1,2,4,6,3,5]=>7
[1,2,4,6,5,3]=>8
[1,2,5,3,4,6]=>3
[1,2,5,3,6,4]=>8
[1,2,5,4,3,6]=>4
[1,2,5,4,6,3]=>9
[1,2,5,6,3,4]=>7
[1,2,5,6,4,3]=>8
[1,2,6,3,4,5]=>3
[1,2,6,3,5,4]=>4
[1,2,6,4,3,5]=>4
[1,2,6,4,5,3]=>5
[1,2,6,5,3,4]=>8
[1,2,6,5,4,3]=>9
[1,3,2,4,5,6]=>2
[1,3,2,4,6,5]=>7
[1,3,2,5,4,6]=>6
[1,3,2,5,6,4]=>11
[1,3,2,6,4,5]=>6
[1,3,2,6,5,4]=>7
[1,3,4,2,5,6]=>5
[1,3,4,2,6,5]=>10
[1,3,4,5,2,6]=>9
[1,3,4,5,6,2]=>14
[1,3,4,6,2,5]=>9
[1,3,4,6,5,2]=>10
[1,3,5,2,4,6]=>5
[1,3,5,2,6,4]=>10
[1,3,5,4,2,6]=>6
[1,3,5,4,6,2]=>11
[1,3,5,6,2,4]=>9
[1,3,5,6,4,2]=>10
[1,3,6,2,4,5]=>5
[1,3,6,2,5,4]=>6
[1,3,6,4,2,5]=>6
[1,3,6,4,5,2]=>7
[1,3,6,5,2,4]=>10
[1,3,6,5,4,2]=>11
[1,4,2,3,5,6]=>2
[1,4,2,3,6,5]=>7
[1,4,2,5,3,6]=>6
[1,4,2,5,6,3]=>11
[1,4,2,6,3,5]=>6
[1,4,2,6,5,3]=>7
[1,4,3,2,5,6]=>3
[1,4,3,2,6,5]=>8
[1,4,3,5,2,6]=>7
[1,4,3,5,6,2]=>12
[1,4,3,6,2,5]=>7
[1,4,3,6,5,2]=>8
[1,4,5,2,3,6]=>5
[1,4,5,2,6,3]=>10
[1,4,5,3,2,6]=>6
[1,4,5,3,6,2]=>11
[1,4,5,6,2,3]=>9
[1,4,5,6,3,2]=>10
[1,4,6,2,3,5]=>5
[1,4,6,2,5,3]=>6
[1,4,6,3,2,5]=>6
[1,4,6,3,5,2]=>7
[1,4,6,5,2,3]=>10
[1,4,6,5,3,2]=>11
[1,5,2,3,4,6]=>2
[1,5,2,3,6,4]=>7
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>8
[1,5,2,6,3,4]=>6
[1,5,2,6,4,3]=>7
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>8
[1,5,3,4,2,6]=>4
[1,5,3,4,6,2]=>9
[1,5,3,6,2,4]=>7
[1,5,3,6,4,2]=>8
[1,5,4,2,3,6]=>6
[1,5,4,2,6,3]=>11
[1,5,4,3,2,6]=>7
[1,5,4,3,6,2]=>12
[1,5,4,6,2,3]=>10
[1,5,4,6,3,2]=>11
[1,5,6,2,3,4]=>5
[1,5,6,2,4,3]=>6
[1,5,6,3,2,4]=>6
[1,5,6,3,4,2]=>7
[1,5,6,4,2,3]=>7
[1,5,6,4,3,2]=>8
[1,6,2,3,4,5]=>2
[1,6,2,3,5,4]=>3
[1,6,2,4,3,5]=>3
[1,6,2,4,5,3]=>4
[1,6,2,5,3,4]=>7
[1,6,2,5,4,3]=>8
[1,6,3,2,4,5]=>3
[1,6,3,2,5,4]=>4
[1,6,3,4,2,5]=>4
[1,6,3,4,5,2]=>5
[1,6,3,5,2,4]=>8
[1,6,3,5,4,2]=>9
[1,6,4,2,3,5]=>6
[1,6,4,2,5,3]=>7
[1,6,4,3,2,5]=>7
[1,6,4,3,5,2]=>8
[1,6,4,5,2,3]=>11
[1,6,4,5,3,2]=>12
[1,6,5,2,3,4]=>6
[1,6,5,2,4,3]=>7
[1,6,5,3,2,4]=>7
[1,6,5,3,4,2]=>8
[1,6,5,4,2,3]=>8
[1,6,5,4,3,2]=>9
[2,1,3,4,5,6]=>1
[2,1,3,4,6,5]=>6
[2,1,3,5,4,6]=>5
[2,1,3,5,6,4]=>10
[2,1,3,6,4,5]=>5
[2,1,3,6,5,4]=>6
[2,1,4,3,5,6]=>4
[2,1,4,3,6,5]=>9
[2,1,4,5,3,6]=>8
[2,1,4,5,6,3]=>13
[2,1,4,6,3,5]=>8
[2,1,4,6,5,3]=>9
[2,1,5,3,4,6]=>4
[2,1,5,3,6,4]=>9
[2,1,5,4,3,6]=>5
[2,1,5,4,6,3]=>10
[2,1,5,6,3,4]=>8
[2,1,5,6,4,3]=>9
[2,1,6,3,4,5]=>4
[2,1,6,3,5,4]=>5
[2,1,6,4,3,5]=>5
[2,1,6,4,5,3]=>6
[2,1,6,5,3,4]=>9
[2,1,6,5,4,3]=>10
[2,3,1,4,5,6]=>3
[2,3,1,4,6,5]=>8
[2,3,1,5,4,6]=>7
[2,3,1,5,6,4]=>12
[2,3,1,6,4,5]=>7
[2,3,1,6,5,4]=>8
[2,3,4,1,5,6]=>6
[2,3,4,1,6,5]=>11
[2,3,4,5,1,6]=>10
[2,3,4,5,6,1]=>15
[2,3,4,6,1,5]=>10
[2,3,4,6,5,1]=>11
[2,3,5,1,4,6]=>6
[2,3,5,1,6,4]=>11
[2,3,5,4,1,6]=>7
[2,3,5,4,6,1]=>12
[2,3,5,6,1,4]=>10
[2,3,5,6,4,1]=>11
[2,3,6,1,4,5]=>6
[2,3,6,1,5,4]=>7
[2,3,6,4,1,5]=>7
[2,3,6,4,5,1]=>8
[2,3,6,5,1,4]=>11
[2,3,6,5,4,1]=>12
[2,4,1,3,5,6]=>3
[2,4,1,3,6,5]=>8
[2,4,1,5,3,6]=>7
[2,4,1,5,6,3]=>12
[2,4,1,6,3,5]=>7
[2,4,1,6,5,3]=>8
[2,4,3,1,5,6]=>4
[2,4,3,1,6,5]=>9
[2,4,3,5,1,6]=>8
[2,4,3,5,6,1]=>13
[2,4,3,6,1,5]=>8
[2,4,3,6,5,1]=>9
[2,4,5,1,3,6]=>6
[2,4,5,1,6,3]=>11
[2,4,5,3,1,6]=>7
[2,4,5,3,6,1]=>12
[2,4,5,6,1,3]=>10
[2,4,5,6,3,1]=>11
[2,4,6,1,3,5]=>6
[2,4,6,1,5,3]=>7
[2,4,6,3,1,5]=>7
[2,4,6,3,5,1]=>8
[2,4,6,5,1,3]=>11
[2,4,6,5,3,1]=>12
[2,5,1,3,4,6]=>3
[2,5,1,3,6,4]=>8
[2,5,1,4,3,6]=>4
[2,5,1,4,6,3]=>9
[2,5,1,6,3,4]=>7
[2,5,1,6,4,3]=>8
[2,5,3,1,4,6]=>4
[2,5,3,1,6,4]=>9
[2,5,3,4,1,6]=>5
[2,5,3,4,6,1]=>10
[2,5,3,6,1,4]=>8
[2,5,3,6,4,1]=>9
[2,5,4,1,3,6]=>7
[2,5,4,1,6,3]=>12
[2,5,4,3,1,6]=>8
[2,5,4,3,6,1]=>13
[2,5,4,6,1,3]=>11
[2,5,4,6,3,1]=>12
[2,5,6,1,3,4]=>6
[2,5,6,1,4,3]=>7
[2,5,6,3,1,4]=>7
[2,5,6,3,4,1]=>8
[2,5,6,4,1,3]=>8
[2,5,6,4,3,1]=>9
[2,6,1,3,4,5]=>3
[2,6,1,3,5,4]=>4
[2,6,1,4,3,5]=>4
[2,6,1,4,5,3]=>5
[2,6,1,5,3,4]=>8
[2,6,1,5,4,3]=>9
[2,6,3,1,4,5]=>4
[2,6,3,1,5,4]=>5
[2,6,3,4,1,5]=>5
[2,6,3,4,5,1]=>6
[2,6,3,5,1,4]=>9
[2,6,3,5,4,1]=>10
[2,6,4,1,3,5]=>7
[2,6,4,1,5,3]=>8
[2,6,4,3,1,5]=>8
[2,6,4,3,5,1]=>9
[2,6,4,5,1,3]=>12
[2,6,4,5,3,1]=>13
[2,6,5,1,3,4]=>7
[2,6,5,1,4,3]=>8
[2,6,5,3,1,4]=>8
[2,6,5,3,4,1]=>9
[2,6,5,4,1,3]=>9
[2,6,5,4,3,1]=>10
[3,1,2,4,5,6]=>1
[3,1,2,4,6,5]=>6
[3,1,2,5,4,6]=>5
[3,1,2,5,6,4]=>10
[3,1,2,6,4,5]=>5
[3,1,2,6,5,4]=>6
[3,1,4,2,5,6]=>4
[3,1,4,2,6,5]=>9
[3,1,4,5,2,6]=>8
[3,1,4,5,6,2]=>13
[3,1,4,6,2,5]=>8
[3,1,4,6,5,2]=>9
[3,1,5,2,4,6]=>4
[3,1,5,2,6,4]=>9
[3,1,5,4,2,6]=>5
[3,1,5,4,6,2]=>10
[3,1,5,6,2,4]=>8
[3,1,5,6,4,2]=>9
[3,1,6,2,4,5]=>4
[3,1,6,2,5,4]=>5
[3,1,6,4,2,5]=>5
[3,1,6,4,5,2]=>6
[3,1,6,5,2,4]=>9
[3,1,6,5,4,2]=>10
[3,2,1,4,5,6]=>2
[3,2,1,4,6,5]=>7
[3,2,1,5,4,6]=>6
[3,2,1,5,6,4]=>11
[3,2,1,6,4,5]=>6
[3,2,1,6,5,4]=>7
[3,2,4,1,5,6]=>5
[3,2,4,1,6,5]=>10
[3,2,4,5,1,6]=>9
[3,2,4,5,6,1]=>14
[3,2,4,6,1,5]=>9
[3,2,4,6,5,1]=>10
[3,2,5,1,4,6]=>5
[3,2,5,1,6,4]=>10
[3,2,5,4,1,6]=>6
[3,2,5,4,6,1]=>11
[3,2,5,6,1,4]=>9
[3,2,5,6,4,1]=>10
[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]=>7
[3,2,6,5,1,4]=>10
[3,2,6,5,4,1]=>11
[3,4,1,2,5,6]=>3
[3,4,1,2,6,5]=>8
[3,4,1,5,2,6]=>7
[3,4,1,5,6,2]=>12
[3,4,1,6,2,5]=>7
[3,4,1,6,5,2]=>8
[3,4,2,1,5,6]=>4
[3,4,2,1,6,5]=>9
[3,4,2,5,1,6]=>8
[3,4,2,5,6,1]=>13
[3,4,2,6,1,5]=>8
[3,4,2,6,5,1]=>9
[3,4,5,1,2,6]=>6
[3,4,5,1,6,2]=>11
[3,4,5,2,1,6]=>7
[3,4,5,2,6,1]=>12
[3,4,5,6,1,2]=>10
[3,4,5,6,2,1]=>11
[3,4,6,1,2,5]=>6
[3,4,6,1,5,2]=>7
[3,4,6,2,1,5]=>7
[3,4,6,2,5,1]=>8
[3,4,6,5,1,2]=>11
[3,4,6,5,2,1]=>12
[3,5,1,2,4,6]=>3
[3,5,1,2,6,4]=>8
[3,5,1,4,2,6]=>4
[3,5,1,4,6,2]=>9
[3,5,1,6,2,4]=>7
[3,5,1,6,4,2]=>8
[3,5,2,1,4,6]=>4
[3,5,2,1,6,4]=>9
[3,5,2,4,1,6]=>5
[3,5,2,4,6,1]=>10
[3,5,2,6,1,4]=>8
[3,5,2,6,4,1]=>9
[3,5,4,1,2,6]=>7
[3,5,4,1,6,2]=>12
[3,5,4,2,1,6]=>8
[3,5,4,2,6,1]=>13
[3,5,4,6,1,2]=>11
[3,5,4,6,2,1]=>12
[3,5,6,1,2,4]=>6
[3,5,6,1,4,2]=>7
[3,5,6,2,1,4]=>7
[3,5,6,2,4,1]=>8
[3,5,6,4,1,2]=>8
[3,5,6,4,2,1]=>9
[3,6,1,2,4,5]=>3
[3,6,1,2,5,4]=>4
[3,6,1,4,2,5]=>4
[3,6,1,4,5,2]=>5
[3,6,1,5,2,4]=>8
[3,6,1,5,4,2]=>9
[3,6,2,1,4,5]=>4
[3,6,2,1,5,4]=>5
[3,6,2,4,1,5]=>5
[3,6,2,4,5,1]=>6
[3,6,2,5,1,4]=>9
[3,6,2,5,4,1]=>10
[3,6,4,1,2,5]=>7
[3,6,4,1,5,2]=>8
[3,6,4,2,1,5]=>8
[3,6,4,2,5,1]=>9
[3,6,4,5,1,2]=>12
[3,6,4,5,2,1]=>13
[3,6,5,1,2,4]=>7
[3,6,5,1,4,2]=>8
[3,6,5,2,1,4]=>8
[3,6,5,2,4,1]=>9
[3,6,5,4,1,2]=>9
[3,6,5,4,2,1]=>10
[4,1,2,3,5,6]=>1
[4,1,2,3,6,5]=>6
[4,1,2,5,3,6]=>5
[4,1,2,5,6,3]=>10
[4,1,2,6,3,5]=>5
[4,1,2,6,5,3]=>6
[4,1,3,2,5,6]=>2
[4,1,3,2,6,5]=>7
[4,1,3,5,2,6]=>6
[4,1,3,5,6,2]=>11
[4,1,3,6,2,5]=>6
[4,1,3,6,5,2]=>7
[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]=>10
[4,1,5,6,2,3]=>8
[4,1,5,6,3,2]=>9
[4,1,6,2,3,5]=>4
[4,1,6,2,5,3]=>5
[4,1,6,3,2,5]=>5
[4,1,6,3,5,2]=>6
[4,1,6,5,2,3]=>9
[4,1,6,5,3,2]=>10
[4,2,1,3,5,6]=>2
[4,2,1,3,6,5]=>7
[4,2,1,5,3,6]=>6
[4,2,1,5,6,3]=>11
[4,2,1,6,3,5]=>6
[4,2,1,6,5,3]=>7
[4,2,3,1,5,6]=>3
[4,2,3,1,6,5]=>8
[4,2,3,5,1,6]=>7
[4,2,3,5,6,1]=>12
[4,2,3,6,1,5]=>7
[4,2,3,6,5,1]=>8
[4,2,5,1,3,6]=>5
[4,2,5,1,6,3]=>10
[4,2,5,3,1,6]=>6
[4,2,5,3,6,1]=>11
[4,2,5,6,1,3]=>9
[4,2,5,6,3,1]=>10
[4,2,6,1,3,5]=>5
[4,2,6,1,5,3]=>6
[4,2,6,3,1,5]=>6
[4,2,6,3,5,1]=>7
[4,2,6,5,1,3]=>10
[4,2,6,5,3,1]=>11
[4,3,1,2,5,6]=>4
[4,3,1,2,6,5]=>9
[4,3,1,5,2,6]=>8
[4,3,1,5,6,2]=>13
[4,3,1,6,2,5]=>8
[4,3,1,6,5,2]=>9
[4,3,2,1,5,6]=>5
[4,3,2,1,6,5]=>10
[4,3,2,5,1,6]=>9
[4,3,2,5,6,1]=>14
[4,3,2,6,1,5]=>9
[4,3,2,6,5,1]=>10
[4,3,5,1,2,6]=>7
[4,3,5,1,6,2]=>12
[4,3,5,2,1,6]=>8
[4,3,5,2,6,1]=>13
[4,3,5,6,1,2]=>11
[4,3,5,6,2,1]=>12
[4,3,6,1,2,5]=>7
[4,3,6,1,5,2]=>8
[4,3,6,2,1,5]=>8
[4,3,6,2,5,1]=>9
[4,3,6,5,1,2]=>12
[4,3,6,5,2,1]=>13
[4,5,1,2,3,6]=>3
[4,5,1,2,6,3]=>8
[4,5,1,3,2,6]=>4
[4,5,1,3,6,2]=>9
[4,5,1,6,2,3]=>7
[4,5,1,6,3,2]=>8
[4,5,2,1,3,6]=>4
[4,5,2,1,6,3]=>9
[4,5,2,3,1,6]=>5
[4,5,2,3,6,1]=>10
[4,5,2,6,1,3]=>8
[4,5,2,6,3,1]=>9
[4,5,3,1,2,6]=>5
[4,5,3,1,6,2]=>10
[4,5,3,2,1,6]=>6
[4,5,3,2,6,1]=>11
[4,5,3,6,1,2]=>9
[4,5,3,6,2,1]=>10
[4,5,6,1,2,3]=>6
[4,5,6,1,3,2]=>7
[4,5,6,2,1,3]=>7
[4,5,6,2,3,1]=>8
[4,5,6,3,1,2]=>8
[4,5,6,3,2,1]=>9
[4,6,1,2,3,5]=>3
[4,6,1,2,5,3]=>4
[4,6,1,3,2,5]=>4
[4,6,1,3,5,2]=>5
[4,6,1,5,2,3]=>8
[4,6,1,5,3,2]=>9
[4,6,2,1,3,5]=>4
[4,6,2,1,5,3]=>5
[4,6,2,3,1,5]=>5
[4,6,2,3,5,1]=>6
[4,6,2,5,1,3]=>9
[4,6,2,5,3,1]=>10
[4,6,3,1,2,5]=>5
[4,6,3,1,5,2]=>6
[4,6,3,2,1,5]=>6
[4,6,3,2,5,1]=>7
[4,6,3,5,1,2]=>10
[4,6,3,5,2,1]=>11
[4,6,5,1,2,3]=>7
[4,6,5,1,3,2]=>8
[4,6,5,2,1,3]=>8
[4,6,5,2,3,1]=>9
[4,6,5,3,1,2]=>9
[4,6,5,3,2,1]=>10
[5,1,2,3,4,6]=>1
[5,1,2,3,6,4]=>6
[5,1,2,4,3,6]=>2
[5,1,2,4,6,3]=>7
[5,1,2,6,3,4]=>5
[5,1,2,6,4,3]=>6
[5,1,3,2,4,6]=>2
[5,1,3,2,6,4]=>7
[5,1,3,4,2,6]=>3
[5,1,3,4,6,2]=>8
[5,1,3,6,2,4]=>6
[5,1,3,6,4,2]=>7
[5,1,4,2,3,6]=>5
[5,1,4,2,6,3]=>10
[5,1,4,3,2,6]=>6
[5,1,4,3,6,2]=>11
[5,1,4,6,2,3]=>9
[5,1,4,6,3,2]=>10
[5,1,6,2,3,4]=>4
[5,1,6,2,4,3]=>5
[5,1,6,3,2,4]=>5
[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]=>2
[5,2,1,3,6,4]=>7
[5,2,1,4,3,6]=>3
[5,2,1,4,6,3]=>8
[5,2,1,6,3,4]=>6
[5,2,1,6,4,3]=>7
[5,2,3,1,4,6]=>3
[5,2,3,1,6,4]=>8
[5,2,3,4,1,6]=>4
[5,2,3,4,6,1]=>9
[5,2,3,6,1,4]=>7
[5,2,3,6,4,1]=>8
[5,2,4,1,3,6]=>6
[5,2,4,1,6,3]=>11
[5,2,4,3,1,6]=>7
[5,2,4,3,6,1]=>12
[5,2,4,6,1,3]=>10
[5,2,4,6,3,1]=>11
[5,2,6,1,3,4]=>5
[5,2,6,1,4,3]=>6
[5,2,6,3,1,4]=>6
[5,2,6,3,4,1]=>7
[5,2,6,4,1,3]=>7
[5,2,6,4,3,1]=>8
[5,3,1,2,4,6]=>4
[5,3,1,2,6,4]=>9
[5,3,1,4,2,6]=>5
[5,3,1,4,6,2]=>10
[5,3,1,6,2,4]=>8
[5,3,1,6,4,2]=>9
[5,3,2,1,4,6]=>5
[5,3,2,1,6,4]=>10
[5,3,2,4,1,6]=>6
[5,3,2,4,6,1]=>11
[5,3,2,6,1,4]=>9
[5,3,2,6,4,1]=>10
[5,3,4,1,2,6]=>8
[5,3,4,1,6,2]=>13
[5,3,4,2,1,6]=>9
[5,3,4,2,6,1]=>14
[5,3,4,6,1,2]=>12
[5,3,4,6,2,1]=>13
[5,3,6,1,2,4]=>7
[5,3,6,1,4,2]=>8
[5,3,6,2,1,4]=>8
[5,3,6,2,4,1]=>9
[5,3,6,4,1,2]=>9
[5,3,6,4,2,1]=>10
[5,4,1,2,3,6]=>4
[5,4,1,2,6,3]=>9
[5,4,1,3,2,6]=>5
[5,4,1,3,6,2]=>10
[5,4,1,6,2,3]=>8
[5,4,1,6,3,2]=>9
[5,4,2,1,3,6]=>5
[5,4,2,1,6,3]=>10
[5,4,2,3,1,6]=>6
[5,4,2,3,6,1]=>11
[5,4,2,6,1,3]=>9
[5,4,2,6,3,1]=>10
[5,4,3,1,2,6]=>6
[5,4,3,1,6,2]=>11
[5,4,3,2,1,6]=>7
[5,4,3,2,6,1]=>12
[5,4,3,6,1,2]=>10
[5,4,3,6,2,1]=>11
[5,4,6,1,2,3]=>7
[5,4,6,1,3,2]=>8
[5,4,6,2,1,3]=>8
[5,4,6,2,3,1]=>9
[5,4,6,3,1,2]=>9
[5,4,6,3,2,1]=>10
[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]=>5
[5,6,1,4,2,3]=>5
[5,6,1,4,3,2]=>6
[5,6,2,1,3,4]=>4
[5,6,2,1,4,3]=>5
[5,6,2,3,1,4]=>5
[5,6,2,3,4,1]=>6
[5,6,2,4,1,3]=>6
[5,6,2,4,3,1]=>7
[5,6,3,1,2,4]=>5
[5,6,3,1,4,2]=>6
[5,6,3,2,1,4]=>6
[5,6,3,2,4,1]=>7
[5,6,3,4,1,2]=>7
[5,6,3,4,2,1]=>8
[5,6,4,1,2,3]=>8
[5,6,4,1,3,2]=>9
[5,6,4,2,1,3]=>9
[5,6,4,2,3,1]=>10
[5,6,4,3,1,2]=>10
[5,6,4,3,2,1]=>11
[6,1,2,3,4,5]=>1
[6,1,2,3,5,4]=>2
[6,1,2,4,3,5]=>2
[6,1,2,4,5,3]=>3
[6,1,2,5,3,4]=>6
[6,1,2,5,4,3]=>7
[6,1,3,2,4,5]=>2
[6,1,3,2,5,4]=>3
[6,1,3,4,2,5]=>3
[6,1,3,4,5,2]=>4
[6,1,3,5,2,4]=>7
[6,1,3,5,4,2]=>8
[6,1,4,2,3,5]=>5
[6,1,4,2,5,3]=>6
[6,1,4,3,2,5]=>6
[6,1,4,3,5,2]=>7
[6,1,4,5,2,3]=>10
[6,1,4,5,3,2]=>11
[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]=>7
[6,1,5,4,2,3]=>7
[6,1,5,4,3,2]=>8
[6,2,1,3,4,5]=>2
[6,2,1,3,5,4]=>3
[6,2,1,4,3,5]=>3
[6,2,1,4,5,3]=>4
[6,2,1,5,3,4]=>7
[6,2,1,5,4,3]=>8
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>4
[6,2,3,4,1,5]=>4
[6,2,3,4,5,1]=>5
[6,2,3,5,1,4]=>8
[6,2,3,5,4,1]=>9
[6,2,4,1,3,5]=>6
[6,2,4,1,5,3]=>7
[6,2,4,3,1,5]=>7
[6,2,4,3,5,1]=>8
[6,2,4,5,1,3]=>11
[6,2,4,5,3,1]=>12
[6,2,5,1,3,4]=>6
[6,2,5,1,4,3]=>7
[6,2,5,3,1,4]=>7
[6,2,5,3,4,1]=>8
[6,2,5,4,1,3]=>8
[6,2,5,4,3,1]=>9
[6,3,1,2,4,5]=>4
[6,3,1,2,5,4]=>5
[6,3,1,4,2,5]=>5
[6,3,1,4,5,2]=>6
[6,3,1,5,2,4]=>9
[6,3,1,5,4,2]=>10
[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]=>7
[6,3,2,5,1,4]=>10
[6,3,2,5,4,1]=>11
[6,3,4,1,2,5]=>8
[6,3,4,1,5,2]=>9
[6,3,4,2,1,5]=>9
[6,3,4,2,5,1]=>10
[6,3,4,5,1,2]=>13
[6,3,4,5,2,1]=>14
[6,3,5,1,2,4]=>8
[6,3,5,1,4,2]=>9
[6,3,5,2,1,4]=>9
[6,3,5,2,4,1]=>10
[6,3,5,4,1,2]=>10
[6,3,5,4,2,1]=>11
[6,4,1,2,3,5]=>4
[6,4,1,2,5,3]=>5
[6,4,1,3,2,5]=>5
[6,4,1,3,5,2]=>6
[6,4,1,5,2,3]=>9
[6,4,1,5,3,2]=>10
[6,4,2,1,3,5]=>5
[6,4,2,1,5,3]=>6
[6,4,2,3,1,5]=>6
[6,4,2,3,5,1]=>7
[6,4,2,5,1,3]=>10
[6,4,2,5,3,1]=>11
[6,4,3,1,2,5]=>6
[6,4,3,1,5,2]=>7
[6,4,3,2,1,5]=>7
[6,4,3,2,5,1]=>8
[6,4,3,5,1,2]=>11
[6,4,3,5,2,1]=>12
[6,4,5,1,2,3]=>8
[6,4,5,1,3,2]=>9
[6,4,5,2,1,3]=>9
[6,4,5,2,3,1]=>10
[6,4,5,3,1,2]=>10
[6,4,5,3,2,1]=>11
[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]=>6
[6,5,1,4,2,3]=>6
[6,5,1,4,3,2]=>7
[6,5,2,1,3,4]=>5
[6,5,2,1,4,3]=>6
[6,5,2,3,1,4]=>6
[6,5,2,3,4,1]=>7
[6,5,2,4,1,3]=>7
[6,5,2,4,3,1]=>8
[6,5,3,1,2,4]=>6
[6,5,3,1,4,2]=>7
[6,5,3,2,1,4]=>7
[6,5,3,2,4,1]=>8
[6,5,3,4,1,2]=>8
[6,5,3,4,2,1]=>9
[6,5,4,1,2,3]=>9
[6,5,4,1,3,2]=>10
[6,5,4,2,1,3]=>10
[6,5,4,2,3,1]=>11
[6,5,4,3,1,2]=>11
[6,5,4,3,2,1]=>12
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Description
The Denert index of a permutation.
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
References
[1] Denert, M. The genus zeta function of hereditary orders in central simple algebras over global fields MathSciNet:0993928
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian MathSciNet:1061147
[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). OEIS:A008302
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian MathSciNet:1061147
[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). OEIS:A008302
Code
import itertools
def statistic(pi):
return sum(1 for l, k in itertools.combinations(range(pi.size()), 2)
if (pi[k] < pi[l] <= k+1)
or (pi[l] <= k+1 < pi[k])
or (k+1 < pi[k] < pi[l]))
Created
Jul 24, 2013 at 12:25 by Christian Stump
Updated
Feb 27, 2023 at 12:37 by Martin Rubey
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