Identifier
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => 1
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 3
[.,[[.,.],.]] => [2,3,1] => [1,3,2] => 2
[[.,.],[.,.]] => [1,3,2] => [3,1,2] => 1
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 6
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,4,3,2] => 5
[.,[[.,.],[.,.]]] => [2,4,3,1] => [4,1,3,2] => 4
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,4,3] => 4
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,4,3] => 3
[[.,.],[.,[.,.]]] => [1,4,3,2] => [4,3,1,2] => 3
[[.,.],[[.,.],.]] => [1,3,4,2] => [2,4,1,3] => 2
[[.,[.,.]],[.,.]] => [2,1,4,3] => [1,4,2,3] => 2
[[[.,.],.],[.,.]] => [1,2,4,3] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 3
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,2,4] => 2
[[[.,.],[.,.]],.] => [1,3,2,4] => [3,1,2,4] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,5,4,3,2] => 9
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [5,1,4,3,2] => 8
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,1,5,4,3] => 8
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,5,4,3] => 7
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [5,4,1,3,2] => 7
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [2,5,1,4,3] => 6
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,5,2,4,3] => 6
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [5,1,2,4,3] => 5
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [3,2,1,5,4] => 7
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,3,2,5,4] => 6
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [3,1,2,5,4] => 5
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,1,3,5,4] => 5
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,5,4] => 4
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [5,4,3,1,2] => 6
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [2,5,4,1,3] => 5
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [5,2,4,1,3] => 4
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [3,2,5,1,4] => 4
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [2,3,5,1,4] => 3
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [5,1,4,2,3] => 4
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [1,3,5,2,4] => 3
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [3,5,1,2,4] => 2
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [2,1,5,3,4] => 4
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [1,2,5,3,4] => 3
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,5,1,3,4] => 2
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [1,5,2,3,4] => 2
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 6
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,3,2,5] => 5
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [4,1,3,2,5] => 4
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [2,1,4,3,5] => 4
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,2,4,3,5] => 3
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [4,3,1,2,5] => 3
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [2,4,1,3,5] => 2
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,4,2,3,5] => 2
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 3
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,2,4,5] => 2
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [3,1,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 15
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,6,5,4,3,2] => 14
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [6,1,5,4,3,2] => 13
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,1,6,5,4,3] => 13
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,6,5,4,3] => 12
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [6,5,1,4,3,2] => 12
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [2,6,1,5,4,3] => 11
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [1,6,2,5,4,3] => 11
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [6,1,2,5,4,3] => 10
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [3,2,1,6,5,4] => 12
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,3,2,6,5,4] => 11
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [3,1,2,6,5,4] => 10
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [2,1,3,6,5,4] => 10
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,3,6,5,4] => 9
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [6,5,4,1,3,2] => 11
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [2,6,5,1,4,3] => 10
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [6,2,5,1,4,3] => 9
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [3,2,6,1,5,4] => 9
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [2,3,6,1,5,4] => 8
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [6,1,5,2,4,3] => 9
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [1,3,6,2,5,4] => 8
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [6,5,1,2,4,3] => 8
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [3,6,1,2,5,4] => 7
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [2,1,6,3,5,4] => 9
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [1,2,6,3,5,4] => 8
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [2,6,1,3,5,4] => 7
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,6,2,3,5,4] => 7
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [6,1,2,3,5,4] => 6
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [4,3,2,1,6,5] => 11
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,4,3,2,6,5] => 10
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [4,1,3,2,6,5] => 9
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [2,1,4,3,6,5] => 9
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,2,4,3,6,5] => 8
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [4,3,1,2,6,5] => 8
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [2,4,1,3,6,5] => 7
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,4,2,3,6,5] => 7
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [4,1,2,3,6,5] => 6
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [3,2,1,4,6,5] => 8
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [1,3,2,4,6,5] => 7
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => [3,1,2,4,6,5] => 6
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [2,1,3,4,6,5] => 6
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 5
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => 10
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => [2,6,5,4,1,3] => 9
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => [6,2,5,4,1,3] => 8
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [3,2,6,5,1,4] => 8
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => [2,3,6,5,1,4] => 7
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => [6,5,2,4,1,3] => 7
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => [3,6,2,5,1,4] => 6
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => 6
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => [6,2,3,5,1,4] => 5
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => [4,3,2,6,1,5] => 7
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => [2,4,3,6,1,5] => 6
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => [4,2,3,6,1,5] => 5
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => [3,2,4,6,1,5] => 5
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => 4
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => [6,5,1,4,2,3] => 7
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => [3,6,1,5,2,4] => 6
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => [6,1,3,5,2,4] => 5
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [4,1,3,6,2,5] => 5
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => [1,3,4,6,2,5] => 4
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 6
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => [3,6,5,1,2,4] => 5
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => [6,3,5,1,2,4] => 4
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => [4,3,6,1,2,5] => 4
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => 3
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => [1,6,2,5,3,4] => 6
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => [2,1,4,6,3,5] => 5
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => [6,1,2,5,3,4] => 5
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => [1,2,4,6,3,5] => 4
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => 4
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => [2,4,6,1,3,5] => 3
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => [6,1,5,2,3,4] => 4
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => [1,4,6,2,3,5] => 3
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => 3
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => [4,6,1,2,3,5] => 2
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => [3,2,1,6,4,5] => 7
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => [1,3,2,6,4,5] => 6
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => [3,1,2,6,4,5] => 5
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => [2,1,3,6,4,5] => 5
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => [1,2,3,6,4,5] => 4
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => 4
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => [2,3,6,1,4,5] => 3
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => [1,3,6,2,4,5] => 3
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => [3,6,1,2,4,5] => 2
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => [2,1,6,3,4,5] => 4
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => [1,2,6,3,4,5] => 3
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => [2,6,1,3,4,5] => 2
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => [1,6,2,3,4,5] => 2
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 10
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [1,5,4,3,2,6] => 9
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => [5,1,4,3,2,6] => 8
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [2,1,5,4,3,6] => 8
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [1,2,5,4,3,6] => 7
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => [5,4,1,3,2,6] => 7
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => [2,5,1,4,3,6] => 6
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => [1,5,2,4,3,6] => 6
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => [5,1,2,4,3,6] => 5
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [3,2,1,5,4,6] => 7
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [1,3,2,5,4,6] => 6
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => [3,1,2,5,4,6] => 5
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [2,1,3,5,4,6] => 5
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [1,2,3,5,4,6] => 4
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => [5,4,3,1,2,6] => 6
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => [2,5,4,1,3,6] => 5
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => [5,2,4,1,3,6] => 4
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => [3,2,5,1,4,6] => 4
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => [2,3,5,1,4,6] => 3
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => [5,1,4,2,3,6] => 4
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => [1,3,5,2,4,6] => 3
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => [5,4,1,2,3,6] => 3
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => [3,5,1,2,4,6] => 2
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [2,1,5,3,4,6] => 4
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => [1,2,5,3,4,6] => 3
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => [2,5,1,3,4,6] => 2
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => [1,5,2,3,4,6] => 2
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 6
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [1,4,3,2,5,6] => 5
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => [4,1,3,2,5,6] => 4
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [2,1,4,3,5,6] => 4
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [1,2,4,3,5,6] => 3
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => 3
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => [2,4,1,3,5,6] => 2
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => [1,4,2,3,5,6] => 2
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => 1
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 3
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [1,3,2,4,5,6] => 2
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 1
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
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
click to show known generating functions       
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.
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 $\sigma$.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.