Identifier
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]] => [2,3,1] => [3,2,1] => 0
[[.,.],[.,.]] => [3,1,2] => [3,1,2] => 0
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [2,3,4,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [2,4,3,1] => 0
[.,[[.,.],[.,.]]] => [4,2,3,1] => [3,4,2,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,3,2,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [4,2,3,1] => 1
[[.,.],[.,[.,.]]] => [4,3,1,2] => [3,1,4,2] => 0
[[.,.],[[.,.],.]] => [3,4,1,2] => [4,1,3,2] => 0
[[.,[.,.]],[.,.]] => [4,2,1,3] => [2,4,1,3] => 0
[[[.,.],.],[.,.]] => [4,1,2,3] => [4,1,2,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.] => [3,1,2,4] => [3,1,2,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [2,3,5,4,1] => 0
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [2,4,5,3,1] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,5,4,3,1] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [2,5,3,4,1] => 1
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [3,4,2,5,1] => 0
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [3,5,2,4,1] => 1
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [4,3,5,2,1] => 0
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [4,5,2,3,1] => 2
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,3,4,2,1] => 2
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [5,4,3,2,1] => 0
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [5,4,2,3,1] => 2
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [5,3,2,4,1] => 2
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [5,2,3,4,1] => 3
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [3,1,4,5,2] => 0
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [3,1,5,4,2] => 0
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [4,1,5,3,2] => 0
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [5,1,4,3,2] => 0
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [5,1,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [2,4,1,5,3] => 0
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [2,5,1,4,3] => 0
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [4,1,2,5,3] => 0
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [5,1,2,4,3] => 0
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [2,3,5,1,4] => 0
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [3,5,2,1,4] => 0
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [3,1,5,2,4] => 0
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [2,5,1,3,4] => 0
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [2,4,3,1,5] => 0
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [3,4,2,1,5] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [4,3,2,1,5] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [4,1,3,2,5] => 0
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [3,2,1,4,5] => 0
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [2,3,4,5,6,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,3,4,6,5,1] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [2,3,5,6,4,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,3,6,5,4,1] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [2,3,6,4,5,1] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [2,4,5,3,6,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [2,4,6,3,5,1] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [2,5,4,6,3,1] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [2,5,6,3,4,1] => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [2,6,4,5,3,1] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [2,6,5,4,3,1] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [2,6,5,3,4,1] => 2
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [2,6,4,3,5,1] => 2
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [2,6,3,4,5,1] => 3
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [3,4,2,5,6,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [3,4,2,6,5,1] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [3,5,2,6,4,1] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [3,6,2,5,4,1] => 2
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [3,6,2,4,5,1] => 3
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [4,3,5,2,6,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [4,3,6,2,5,1] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [4,5,2,3,6,1] => 2
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [4,6,2,3,5,1] => 4
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [5,3,4,6,2,1] => 2
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [5,4,6,3,2,1] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [5,4,2,6,3,1] => 2
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [5,3,6,2,4,1] => 3
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [5,6,2,3,4,1] => 6
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,3,4,5,2,1] => 6
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [6,3,5,4,2,1] => 4
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [6,4,5,3,2,1] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [6,5,4,3,2,1] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [6,5,3,4,2,1] => 4
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [6,4,2,5,3,1] => 5
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [6,5,2,4,3,1] => 4
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [6,3,5,2,4,1] => 5
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [6,5,2,3,4,1] => 6
>>> Load all 314 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [6,3,4,2,5,1] => 5
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [6,4,3,2,5,1] => 3
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => [6,4,2,3,5,1] => 5
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [6,3,2,4,5,1] => 5
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 6
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => [3,1,4,5,6,2] => 0
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => [3,1,4,6,5,2] => 0
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => [3,1,5,6,4,2] => 0
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => [3,1,6,5,4,2] => 0
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => [3,1,6,4,5,2] => 1
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => [4,1,5,3,6,2] => 0
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => [4,1,6,3,5,2] => 1
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => [5,1,4,6,3,2] => 0
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => [5,1,6,3,4,2] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => [6,1,4,5,3,2] => 2
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => [6,1,5,4,3,2] => 0
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => [6,1,5,3,4,2] => 2
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => [6,1,4,3,5,2] => 2
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => [6,1,3,4,5,2] => 3
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => [2,4,1,5,6,3] => 0
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => [2,4,1,6,5,3] => 0
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => [2,5,1,6,4,3] => 0
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => [2,6,1,5,4,3] => 0
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => [2,6,1,4,5,3] => 1
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => [4,1,2,5,6,3] => 0
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => [4,1,2,6,5,3] => 0
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => [5,1,2,6,4,3] => 0
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => [6,1,2,5,4,3] => 0
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => [6,1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => [2,3,5,1,6,4] => 0
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => [2,3,6,1,5,4] => 0
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => [3,5,2,1,6,4] => 0
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => [3,6,2,1,5,4] => 0
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => [3,1,5,2,6,4] => 0
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => [3,1,6,2,5,4] => 0
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => [2,5,1,3,6,4] => 0
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => [2,6,1,3,5,4] => 0
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => [5,1,2,3,6,4] => 0
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => [6,1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => [2,3,4,6,1,5] => 0
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => [2,4,6,3,1,5] => 0
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => [3,4,2,6,1,5] => 0
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => [4,3,6,2,1,5] => 0
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => [4,6,2,3,1,5] => 2
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => [3,1,4,6,2,5] => 0
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 0
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => [2,4,1,6,3,5] => 0
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => [4,1,2,6,3,5] => 0
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => [2,3,6,1,4,5] => 0
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => [3,6,2,1,4,5] => 0
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 0
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => [2,6,1,3,4,5] => 0
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [2,3,4,5,1,6] => 0
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [2,3,5,4,1,6] => 0
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => [2,4,5,3,1,6] => 0
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [2,5,4,3,1,6] => 0
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [2,5,3,4,1,6] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => [3,4,2,5,1,6] => 0
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => [3,5,2,4,1,6] => 1
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => [4,3,5,2,1,6] => 0
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => [4,5,2,3,1,6] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [5,3,4,2,1,6] => 2
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [5,4,3,2,1,6] => 0
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => [5,4,2,3,1,6] => 2
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [5,3,2,4,1,6] => 2
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => 3
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => [3,1,4,5,2,6] => 0
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => [3,1,5,4,2,6] => 0
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => [4,1,5,3,2,6] => 0
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => [5,1,4,3,2,6] => 0
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => [5,1,3,4,2,6] => 1
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => [2,4,1,5,3,6] => 0
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => [2,5,1,4,3,6] => 0
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => [4,1,2,5,3,6] => 0
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => [5,1,2,4,3,6] => 0
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => [2,3,5,1,4,6] => 0
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => [3,5,2,1,4,6] => 0
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => [3,1,5,2,4,6] => 0
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => [2,5,1,3,4,6] => 0
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => 0
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [2,3,4,1,5,6] => 0
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [2,4,3,1,5,6] => 0
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => [3,4,2,1,5,6] => 0
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [4,3,2,1,5,6] => 0
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [4,2,3,1,5,6] => 1
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => [3,1,4,2,5,6] => 0
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => [4,1,3,2,5,6] => 0
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 0
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => 0
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [2,3,1,4,5,6] => 0
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 0
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => 0
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]] => [7,6,5,4,2,3,1] => [3,4,2,5,6,7,1] => 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]] => [7,6,5,3,2,4,1] => [4,3,5,2,6,7,1] => 0
[.,[[[.,.],.],[.,[.,[.,.]]]]] => [7,6,5,2,3,4,1] => [4,5,2,3,6,7,1] => 2
[.,[[.,[[.,[[.,.],.]],.]],.]] => [4,5,3,6,2,7,1] => [7,6,5,4,3,2,1] => 0
[[[[.,.],.],.],[.,[.,[.,.]]]] => [7,6,5,1,2,3,4] => [5,1,2,3,6,7,4] => 0
[[.,[.,[.,[.,.]]]],[.,[.,.]]] => [7,6,4,3,2,1,5] => [2,3,4,6,1,7,5] => 0
[[[[[.,.],.],.],.],[.,[.,.]]] => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => 0
[[.,[.,[.,[.,[.,.]]]]],[.,.]] => [7,5,4,3,2,1,6] => [2,3,4,5,7,1,6] => 0
[[[[[.,.],.],.],[.,.]],[.,.]] => [7,5,1,2,3,4,6] => [5,1,2,3,7,4,6] => 0
[[[.,[.,[.,[.,.]]]],.],[.,.]] => [7,4,3,2,1,5,6] => [2,3,4,7,1,5,6] => 0
[[[[[.,[.,.]],.],.],.],[.,.]] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => 0
[[[[[[.,.],.],.],.],.],[.,.]] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 0
[[.,[.,[.,[.,[.,[.,.]]]]]],.] => [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => 0
[[.,[.,[[.,[[.,.],.]],.]]],.] => [4,5,3,6,2,1,7] => [2,6,5,4,3,1,7] => 0
[[.,[[.,[[.,.],.]],[.,.]]],.] => [6,3,4,2,5,1,7] => [5,4,6,3,2,1,7] => 0
[[.,[[.,[.,[.,[.,.]]]],.]],.] => [5,4,3,2,6,1,7] => [6,3,4,5,2,1,7] => 6
[[.,[[.,[.,[[.,.],.]]],.]],.] => [4,5,3,2,6,1,7] => [6,3,5,4,2,1,7] => 4
[[.,[[.,[[.,[.,.]],.]],.]],.] => [4,3,5,2,6,1,7] => [6,5,4,3,2,1,7] => 0
[[.,[[.,[[[.,.],.],.]],.]],.] => [3,4,5,2,6,1,7] => [6,5,3,4,2,1,7] => 4
[[.,[[[.,[.,[.,.]]],.],.]],.] => [4,3,2,5,6,1,7] => [6,3,4,2,5,1,7] => 5
[[.,[[[.,[[.,.],.]],.],.]],.] => [3,4,2,5,6,1,7] => [6,4,3,2,5,1,7] => 3
[[[[[.,.],.],.],[.,[.,.]]],.] => [6,5,1,2,3,4,7] => [5,1,2,3,6,4,7] => 0
[[[[[.,[.,.]],.],.],[.,.]],.] => [6,2,1,3,4,5,7] => [2,6,1,3,4,5,7] => 0
[[[[[[.,.],.],.],.],[.,.]],.] => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => 0
[[[.,[.,[.,[.,[.,.]]]]],.],.] => [5,4,3,2,1,6,7] => [2,3,4,5,1,6,7] => 0
[[[.,[[.,[[.,.],.]],.]],.],.] => [3,4,2,5,1,6,7] => [5,4,3,2,1,6,7] => 0
[[[[[.,[.,.]],.],[.,.]],.],.] => [5,2,1,3,4,6,7] => [2,5,1,3,4,6,7] => 0
[[[[[[.,.],.],.],[.,.]],.],.] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => 0
[[[[.,[.,[.,[.,.]]]],.],.],.] => [4,3,2,1,5,6,7] => [2,3,4,1,5,6,7] => 0
[[[[.,[[.,[.,.]],.]],.],.],.] => [3,2,4,1,5,6,7] => [4,3,2,1,5,6,7] => 0
[[[[[.,[.,.]],[.,.]],.],.],.] => [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => 0
[[[[[[.,.],.],[.,.]],.],.],.] => [4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => 0
[[[[[.,[.,[.,.]]],.],.],.],.] => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => 0
[[[[[.,[[.,.],.]],.],.],.],.] => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => 0
[[[[[[.,.],[.,.]],.],.],.],.] => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => 0
[[[[[[.,[.,.]],.],.],.],.],.] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,1] => 0
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]] => [5,6,7,8,4,3,2,1] => [2,3,4,8,5,6,7,1] => 3
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]] => [7,4,5,6,8,3,2,1] => [2,3,8,7,4,5,6,1] => 6
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]] => [8,7,6,5,4,2,3,1] => [3,4,2,5,6,7,8,1] => 0
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]] => [8,7,5,6,3,2,4,1] => [4,3,6,2,7,5,8,1] => 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]] => [5,4,6,3,7,2,8,1] => [8,7,6,5,4,3,2,1] => 0
[.,[[[[.,[.,.]],.],[.,[.,.]]],.]] => [7,6,3,2,4,5,8,1] => [8,3,6,2,4,7,5,1] => 15
[.,[[[[.,.],[.,[.,.]]],[.,.]],.]] => [7,5,4,2,3,6,8,1] => [8,4,2,5,7,3,6,1] => 15
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]] => [6,5,7,8,4,3,1,2] => [3,1,4,8,6,5,7,2] => 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]] => [8,7,5,4,6,3,1,2] => [3,1,6,5,7,4,8,2] => 0
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]] => [8,7,5,6,3,4,1,2] => [4,1,6,3,7,5,8,2] => 1
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]] => [6,5,4,7,3,8,1,2] => [8,1,7,5,6,4,3,2] => 6
[[.,.],[[[[.,.],[.,.]],[.,.]],.]] => [7,5,3,4,6,8,1,2] => [8,1,5,3,7,4,6,2] => 9
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]] => [6,7,5,8,4,2,1,3] => [2,4,1,8,7,6,5,3] => 0
[[[.,.],.],[.,[[[.,.],.],[.,.]]]] => [8,5,6,7,4,1,2,3] => [4,1,2,7,8,5,6,3] => 2
[[[.,.],.],[.,[[.,[[.,.],.]],.]]] => [6,7,5,8,4,1,2,3] => [4,1,2,8,7,6,5,3] => 0
[[[.,.],.],[.,[[[.,.],[.,.]],.]]] => [7,5,6,8,4,1,2,3] => [4,1,2,8,7,5,6,3] => 2
[[[.,.],.],[[.,[.,[.,[.,.]]]],.]] => [7,6,5,4,8,1,2,3] => [8,1,2,5,6,7,4,3] => 6
[[[.,.],[.,.]],[[.,.],[.,[.,.]]]] => [8,7,5,6,3,1,2,4] => [3,1,6,2,7,5,8,4] => 0
[[[.,[.,[.,.]]],.],[[[.,.],.],.]] => [6,7,8,3,2,1,4,5] => [2,3,8,1,4,6,7,5] => 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]] => [7,6,8,3,1,2,4,5] => [3,1,8,2,4,7,6,5] => 0
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]] => [8,7,5,3,2,4,1,6] => [4,3,5,2,7,1,8,6] => 0
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]] => [8,7,5,3,4,1,2,6] => [4,1,5,3,7,2,8,6] => 0
[[[[.,.],[.,.]],[.,.]],[.,[.,.]]] => [8,7,5,3,1,2,4,6] => [3,1,5,2,7,4,8,6] => 0
[[[[[.,[.,.]],.],.],.],[[.,.],.]] => [7,8,2,1,3,4,5,6] => [2,8,1,3,4,5,7,6] => 0
[[[[[[.,.],.],.],.],.],[.,[.,.]]] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,8,6] => 0
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]] => [8,6,5,4,3,2,1,7] => [2,3,4,5,6,8,1,7] => 0
[[[.,[[[.,.],.],[.,.]]],.],[.,.]] => [8,5,2,3,4,1,6,7] => [4,5,2,3,8,1,6,7] => 2
[[[[.,.],[[.,[.,.]],.]],.],[.,.]] => [8,4,3,5,1,2,6,7] => [5,1,4,8,3,2,6,7] => 0
[[[[[[.,[.,.]],.],.],.],.],[.,.]] => [8,2,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => 0
[[[[[[[.,.],.],.],.],.],.],[.,.]] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 0
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.] => [7,6,5,4,3,2,1,8] => [2,3,4,5,6,7,1,8] => 0
[[.,[[[.,.],.],[.,[[.,.],.]]]],.] => [6,7,5,2,3,4,1,8] => [4,5,2,3,7,6,1,8] => 2
[[.,[[.,[[.,[.,.]],.]],[.,.]]],.] => [7,4,3,5,2,6,1,8] => [6,5,4,7,3,2,1,8] => 0
[[.,[[.,[.,[[.,[.,.]],.]]],.]],.] => [5,4,6,3,2,7,1,8] => [7,3,6,5,4,2,1,8] => 6
[[.,[[.,[[[.,.],.],[.,.]]],.]],.] => [6,3,4,5,2,7,1,8] => [7,5,6,3,4,2,1,8] => 10
[[.,[[.,[[.,[.,[.,.]]],.]],.]],.] => [5,4,3,6,2,7,1,8] => [7,6,4,5,3,2,1,8] => 6
[[.,[[.,[[.,[[.,.],.]],.]],.]],.] => [4,5,3,6,2,7,1,8] => [7,6,5,4,3,2,1,8] => 0
[[.,[[.,[[[.,[.,.]],.],.]],.]],.] => [4,3,5,6,2,7,1,8] => [7,6,4,3,5,2,1,8] => 8
[[.,[[.,[[[[.,.],.],.],.]],.]],.] => [3,4,5,6,2,7,1,8] => [7,6,3,4,5,2,1,8] => 12
[[[[[[.,[.,.]],.],.],.],[.,.]],.] => [7,2,1,3,4,5,6,8] => [2,7,1,3,4,5,6,8] => 0
[[[[[[[.,.],.],.],.],.],[.,.]],.] => [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6,8] => 0
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.] => [6,5,4,3,2,1,7,8] => [2,3,4,5,6,1,7,8] => 0
[[[.,[[.,[[.,[.,.]],.]],.]],.],.] => [4,3,5,2,6,1,7,8] => [6,5,4,3,2,1,7,8] => 0
[[[[[[.,[.,.]],.],.],[.,.]],.],.] => [6,2,1,3,4,5,7,8] => [2,6,1,3,4,5,7,8] => 0
[[[[[[[.,.],.],.],.],[.,.]],.],.] => [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7,8] => 0
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.] => [5,4,3,2,1,6,7,8] => [2,3,4,5,1,6,7,8] => 0
[[[[.,[[.,[[.,.],.]],.]],.],.],.] => [3,4,2,5,1,6,7,8] => [5,4,3,2,1,6,7,8] => 0
[[[[[[[.,.],.],.],[.,.]],.],.],.] => [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7,8] => 0
[[[[[.,[[.,[.,.]],.]],.],.],.],.] => [3,2,4,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 0
[[[[[[.,[[.,.],.]],.],.],.],.],.] => [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 0
[[[[[[[.,[.,.]],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 0
[[[[[[[[.,.],.],.],.],.],.],.],.] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [9,8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,9,1] => 0
[[[[[[[[[.,.],.],.],.],.],.],.],.],.] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [10,9,8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,9,10,1] => 0
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] => [8,7,6,5,4,3,2,1,9] => [2,3,4,5,6,7,8,1,9] => 0
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.] => [9,8,7,6,5,4,3,2,1,10] => [2,3,4,5,6,7,8,9,1,10] => 0
[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.] => [5,6,4,7,3,8,2,9,1,10] => [9,8,7,6,5,4,3,2,1,10] => 0
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 0
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 0
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]] => [9,2,1,3,4,5,6,7,8] => [2,9,1,3,4,5,6,7,8] => 0
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]] => [11,1,2,3,4,5,6,7,8,9,10] => [11,1,2,3,4,5,6,7,8,9,10] => 0
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]] => [10,2,1,3,4,5,6,7,8,9] => [2,10,1,3,4,5,6,7,8,9] => 0
[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]] => [9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,9,7] => 0
[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]] => [6,5,7,4,8,3,9,2,10,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]] => [9,7,6,5,4,3,2,1,8] => [2,3,4,5,6,7,9,1,8] => 0
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.] => [7,6,5,4,3,2,1,8,9] => [2,3,4,5,6,7,1,8,9] => 0
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 0
[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 0
[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]] => [9,8,7,6,5,4,2,3,1] => [3,4,2,5,6,7,8,9,1] => 0
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => 0
[[[[[[[.,[.,.]],.],.],.],.],[.,.]],.] => [8,2,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7,9] => 0
[[[[[[[.,[[.,.],.]],.],.],.],.],.],.] => [2,3,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 0
[[[[[[[[.,.],.],.],.],.],[.,.]],.],.] => [7,1,2,3,4,5,6,8,9] => [7,1,2,3,4,5,6,8,9] => 0
[[[[[[[[.,.],.],.],.],.],.],[.,.]],.] => [8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7,9] => 0
[[[[[[[[[.,.],.],.],.],.],.],.],[.,.]],.] => [9,1,2,3,4,5,6,7,8,10] => [9,1,2,3,4,5,6,7,8,10] => 0
[[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]] => [10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,10,8] => 0
[[[[[[.,[[.,[.,.]],.]],.],.],.],.],.] => [3,2,4,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 0
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Description
The number of occurrences of the pattern 4231 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to be smooth [2].
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
  • the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
  • the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
  • the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
  • the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
  • the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
Map
to 132-avoiding permutation
Description
Return a 132-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 maximal element of the Sylvester class.