Identifier
-
Mp00034:
Dyck paths
—to binary tree: up step, left tree, down step, right tree⟶
Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤ
Values
[1,0] => [.,.] => [1] => [1] => 0
[1,0,1,0] => [.,[.,.]] => [2,1] => [2,1] => 1
[1,1,0,0] => [[.,.],.] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0] => [.,[[.,.],.]] => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0] => [[.,.],[.,.]] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [[.,[.,.]],.] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0] => [[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => [3,4,2,1] => [4,3,2,1] => 1
[1,0,1,1,0,0,1,0] => [.,[[.,.],[.,.]]] => [2,4,3,1] => [4,3,2,1] => 1
[1,0,1,1,0,1,0,0] => [.,[[.,[.,.]],.]] => [3,2,4,1] => [4,2,3,1] => 1
[1,0,1,1,1,0,0,0] => [.,[[[.,.],.],.]] => [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0] => [[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0] => [[.,.],[[.,.],.]] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,1,0,0,1,0] => [[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,1,0,0] => [[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0] => [[.,[[.,.],.]],.] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0] => [[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0] => [[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0] => [[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,0,1,0] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [5,2,4,3,1] => 1
[1,0,1,1,0,1,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,1,0,0,0] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [5,3,2,4,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [5,2,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [5,3,2,4,1] => 1
[1,0,1,1,1,0,1,0,0,0] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,1,0,0,1,0] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [4,2,3,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,1,0,1,0,0,1,0,0] => [[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [6,5,3,4,2,1] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [6,5,3,4,2,1] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [6,2,5,4,3,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [6,2,5,4,3,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [6,3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [6,3,2,5,4,1] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [6,2,5,4,3,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [6,2,5,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [6,3,2,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [6,4,3,2,5,1] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [6,2,3,5,4,1] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [6,2,4,3,5,1] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [6,3,2,4,5,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [6,3,2,4,5,1] => 1
>>> Load all 741 entries. <<<[1,0,1,1,1,1,0,0,0,0,1,0] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [6,2,3,5,4,1] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [6,2,4,3,5,1] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => [6,3,2,4,5,1] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [6,2,3,4,5,1] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => [1,6,3,5,4,2] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => [1,6,4,3,5,2] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => [1,6,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => [1,6,3,5,4,2] => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => [1,6,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => [1,6,3,4,5,2] => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => [2,1,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [2,1,6,4,5,3] => 2
[1,1,0,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => [2,1,6,4,5,3] => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,1,0,0] => [[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => [3,2,1,6,5,4] => 2
[1,1,0,1,0,1,0,1,0,0,1,0] => [[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => [3,2,1,6,5,4] => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 2
[1,1,0,1,1,0,0,1,0,0,1,0] => [[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => [[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => [5,4,3,2,1,6] => 1
[1,1,0,1,1,0,1,0,0,0,1,0] => [[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => [4,2,3,1,6,5] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => [5,2,4,3,1,6] => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [5,3,2,4,1,6] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [5,3,2,4,1,6] => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => [4,2,3,1,6,5] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => [5,2,4,3,1,6] => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => [5,3,2,4,1,6] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [5,2,3,4,1,6] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,0,1,0] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => [1,2,6,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => [1,2,6,4,5,3] => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => [1,5,4,3,2,6] => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => [1,5,4,3,2,6] => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => [1,5,3,4,2,6] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => [2,1,3,6,5,4] => 2
[1,1,1,0,1,0,0,0,1,1,0,0] => [[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => 2
[1,1,1,0,1,0,0,1,0,0,1,0] => [[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,1,1,0,1,0,0,1,0,1,0,0] => [[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => 2
[1,1,1,0,1,0,0,1,1,0,0,0] => [[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => [3,2,1,4,6,5] => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [4,3,2,1,5,6] => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => [4,3,2,1,5,6] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [4,2,3,1,5,6] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [4,2,3,1,5,6] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => [[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => [6,7,5,4,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[.,[[.,.],[.,.]]]]]] => [5,7,6,4,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[.,[[.,[.,.]],.]]]]] => [6,5,7,4,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[[[.,.],.],.]]]]] => [5,6,7,4,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[.,[[.,.],[.,[.,.]]]]]] => [4,7,6,5,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[.,[[.,.],[[.,.],.]]]]] => [4,6,7,5,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [.,[.,[.,[[.,[.,.]],[.,.]]]]] => [5,4,7,6,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[.,[[.,[.,[.,.]]],.]]]] => [6,5,4,7,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [.,[.,[.,[[.,[[.,.],.]],.]]]] => [5,6,4,7,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[.,[[[.,.],.],[.,.]]]]] => [4,5,7,6,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[.,[[[.,.],[.,.]],.]]]] => [4,6,5,7,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[.,[[[.,[.,.]],.],.]]]] => [5,4,6,7,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[[[[.,.],.],.],.]]]] => [4,5,6,7,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,[.,.]]]]]] => [3,7,6,5,4,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [.,[.,[[.,.],[.,[[.,.],.]]]]] => [3,6,7,5,4,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [.,[.,[[.,.],[[.,.],[.,.]]]]] => [3,5,7,6,4,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [.,[.,[[.,.],[[.,[.,.]],.]]]] => [3,6,5,7,4,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [.,[.,[[.,.],[[[.,.],.],.]]]] => [3,5,6,7,4,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [.,[.,[[.,[.,.]],[.,[.,.]]]]] => [4,3,7,6,5,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => [4,3,6,7,5,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => [3,4,7,6,5,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [.,[.,[[[.,.],.],[[.,.],.]]]] => [3,4,6,7,5,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,[.,.]]]]]] => [2,7,6,5,4,3,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [.,[[.,.],[.,[.,[[.,.],.]]]]] => [2,6,7,5,4,3,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => [2,5,7,6,4,3,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [.,[[.,.],[.,[[.,[.,.]],.]]]] => [2,6,5,7,4,3,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [.,[[.,.],[.,[[[.,.],.],.]]]] => [2,5,6,7,4,3,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [.,[[.,.],[[.,.],[.,[.,.]]]]] => [2,4,7,6,5,3,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [.,[[.,.],[[.,.],[[.,.],.]]]] => [2,4,6,7,5,3,1] => [7,6,5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[.,.],[.,[.,[[.,[.,.]],.]]]] => [1,6,5,7,4,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[.,.],[.,[.,[[[.,.],.],.]]]] => [1,5,6,7,4,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[.,.],[.,[[.,[.,.]],[.,.]]]] => [1,5,4,7,6,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [[.,.],[.,[[.,[.,[.,.]]],.]]] => [1,6,5,4,7,3,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [[.,.],[.,[[.,[[.,.],.]],.]]] => [1,5,6,4,7,3,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[.,.],[.,[[[.,.],.],[.,.]]]] => [1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [[.,.],[.,[[[.,.],[.,.]],.]]] => [1,4,6,5,7,3,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0] => [[.,.],[.,[[[.,[.,.]],.],.]]] => [1,5,4,6,7,3,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[[[[.,.],.],.],.]]] => [1,4,5,6,7,3,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[.,.],[[.,.],[.,[[.,.],.]]]] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[.,.],[[.,.],[[.,.],[.,.]]]] => [1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[.,.],[[.,.],[[.,[.,.]],.]]] => [1,3,6,5,7,4,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[.,.],[[.,.],[[[.,.],.],.]]] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => [1,4,3,7,6,5,2] => [1,7,3,6,5,4,2] => 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[.,.],[[.,[.,.]],[[.,.],.]]] => [1,4,3,6,7,5,2] => [1,7,3,6,5,4,2] => 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[.,.],[[.,[.,[.,.]]],[.,.]]] => [1,5,4,3,7,6,2] => [1,7,4,3,6,5,2] => 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [[.,.],[[.,[.,[.,[.,.]]]],.]] => [1,6,5,4,3,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [[.,.],[[.,[.,[[.,.],.]]],.]] => [1,5,6,4,3,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [[.,.],[[.,[[.,.],.]],[.,.]]] => [1,4,5,3,7,6,2] => [1,7,4,3,6,5,2] => 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [[.,.],[[.,[[.,.],[.,.]]],.]] => [1,4,6,5,3,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0] => [[.,.],[[.,[[.,[.,.]],.]],.]] => [1,5,4,6,3,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [[.,.],[[.,[[[.,.],.],.]],.]] => [1,4,5,6,3,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[.,.],[[[.,.],.],[.,[.,.]]]] => [1,3,4,7,6,5,2] => [1,7,3,6,5,4,2] => 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[.,.],[[[.,.],.],[[.,.],.]]] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [[.,.],[[[.,.],[.,.]],[.,.]]] => [1,3,5,4,7,6,2] => [1,7,4,3,6,5,2] => 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,.],[.,[.,.]]],.]] => [1,3,6,5,4,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [[.,.],[[[.,.],[[.,.],.]],.]] => [1,3,5,6,4,7,2] => [1,7,5,4,3,6,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[.,[.,.]],[.,[.,[.,[.,.]]]]] => [2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[.,[.,.]],[.,[.,[[.,.],.]]]] => [2,1,6,7,5,4,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[.,[.,.]],[.,[[.,.],[.,.]]]] => [2,1,5,7,6,4,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[.,[.,.]],[.,[[.,[.,.]],.]]] => [2,1,6,5,7,4,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[[[.,.],.],.]]] => [2,1,5,6,7,4,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[.,[.,.]],[[.,.],[.,[.,.]]]] => [2,1,4,7,6,5,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[.,[.,.]],[[.,.],[[.,.],.]]] => [2,1,4,6,7,5,3] => [2,1,7,6,5,4,3] => 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[.,[.,[.,[.,[.,.]]]]],[.,.]] => [5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,[.,.]]]]]],.] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[.,[[.,.],.]]]]],.] => [5,6,4,3,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [[.,[.,[.,[[.,.],.]]]],[.,.]] => [4,5,3,2,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [[.,[.,[.,[[.,.],[.,.]]]]],.] => [4,6,5,3,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[.,[.,[.,[[.,[.,.]],.]]]],.] => [5,4,6,3,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[.,[[[.,.],.],.]]]],.] => [4,5,6,3,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [[.,[.,[[.,.],[.,.]]]],[.,.]] => [3,5,4,2,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [[.,[.,[[.,.],[.,[.,.]]]]],.] => [3,6,5,4,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [[.,[.,[[.,.],[[.,.],.]]]],.] => [3,5,6,4,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [[.,[.,[[.,[.,.]],.]]],[.,.]] => [4,3,5,2,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[.,[.,[[.,[.,.]],[.,.]]]],.] => [4,3,6,5,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[.,[.,[[.,[.,[.,.]]],.]]],.] => [5,4,3,6,2,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[.,[.,[[.,[[.,.],.]],.]]],.] => [4,5,3,6,2,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[.,[.,[[[.,.],.],.]]],[.,.]] => [3,4,5,2,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [[.,[.,[[[.,.],.],[.,.]]]],.] => [3,4,6,5,2,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [[.,[.,[[[.,.],[.,.]],.]]],.] => [3,5,4,6,2,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[.,[.,[[[.,[.,.]],.],.]]],.] => [4,3,5,6,2,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [[.,[.,[[[[.,.],.],.],.]]],.] => [3,4,5,6,2,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [[.,[[.,.],[.,[.,.]]]],[.,.]] => [2,5,4,3,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [[.,[[.,.],[.,[.,[.,.]]]]],.] => [2,6,5,4,3,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [[.,[[.,.],[.,[[.,.],.]]]],.] => [2,5,6,4,3,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [[.,[[.,.],[[.,.],.]]],[.,.]] => [2,4,5,3,1,7,6] => [5,4,3,2,1,7,6] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [[.,[[.,.],[[.,.],[.,.]]]],.] => [2,4,6,5,3,1,7] => [6,5,4,3,2,1,7] => 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [[.,[[.,.],[[.,[.,.]],.]]],.] => [2,5,4,6,3,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [[.,[[.,.],[[[.,.],.],.]]],.] => [2,4,5,6,3,1,7] => [6,5,3,4,2,1,7] => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [[.,[[.,[.,.]],[.,[.,.]]]],.] => [3,2,6,5,4,1,7] => [6,2,5,4,3,1,7] => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [[.,[[.,[.,.]],[[.,.],.]]],.] => [3,2,5,6,4,1,7] => [6,2,5,4,3,1,7] => 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [[.,[[.,[.,[.,.]]],[.,.]]],.] => [4,3,2,6,5,1,7] => [6,3,2,5,4,1,7] => 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [[.,[[.,[.,[.,[.,.]]]],.]],.] => [5,4,3,2,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [[.,[[.,[.,[[.,.],.]]],.]],.] => [4,5,3,2,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [[.,[[.,[[.,.],.]],[.,.]]],.] => [3,4,2,6,5,1,7] => [6,3,2,5,4,1,7] => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [[.,[[.,[[.,.],[.,.]]],.]],.] => [3,5,4,2,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [[.,[[.,[[.,[.,.]],.]],.]],.] => [4,3,5,2,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [[.,[[.,[[[.,.],.],.]],.]],.] => [3,4,5,2,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [[.,[[[.,.],.],[.,[.,.]]]],.] => [2,3,6,5,4,1,7] => [6,2,5,4,3,1,7] => 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [[.,[[[.,.],.],[[.,.],.]]],.] => [2,3,5,6,4,1,7] => [6,2,5,4,3,1,7] => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [[.,[[[.,.],[.,.]],[.,.]]],.] => [2,4,3,6,5,1,7] => [6,3,2,5,4,1,7] => 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [[.,[[[.,.],[.,[.,.]]],.]],.] => [2,5,4,3,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [[.,[[[.,.],[[.,.],.]],.]],.] => [2,4,5,3,6,1,7] => [6,4,3,2,5,1,7] => 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,[.,.]]]]] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[[.,.],.],[.,[.,[[.,.],.]]]] => [1,2,6,7,5,4,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[[.,.],.],[.,[[.,.],[.,.]]]] => [1,2,5,7,6,4,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [[[.,.],.],[.,[[.,[.,.]],.]]] => [1,2,6,5,7,4,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[[[.,.],.],.]]] => [1,2,5,6,7,4,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[[.,.],.],[[.,.],[.,[.,.]]]] => [1,2,4,7,6,5,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[.,.],.],[[.,.],[[.,.],.]]] => [1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [[[.,.],.],[[.,[.,.]],[.,.]]] => [1,2,5,4,7,6,3] => [1,2,7,4,6,5,3] => 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [[[.,.],.],[[.,[.,[.,.]]],.]] => [1,2,6,5,4,7,3] => [1,2,7,5,4,6,3] => 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [[[.,.],.],[[.,[[.,.],.]],.]] => [1,2,5,6,4,7,3] => [1,2,7,5,4,6,3] => 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[[.,.],.],[[[.,.],.],[.,.]]] => [1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [[[.,.],.],[[[.,.],[.,.]],.]] => [1,2,4,6,5,7,3] => [1,2,7,5,4,6,3] => 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [[[.,.],.],[[[.,[.,.]],.],.]] => [1,2,5,4,6,7,3] => [1,2,7,4,5,6,3] => 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[[[[.,.],.],.],.]] => [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [[[.,.],[.,.]],[.,[.,[.,.]]]] => [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [[[.,.],[.,.]],[.,[[.,.],.]]] => [1,3,2,6,7,5,4] => [1,3,2,7,6,5,4] => 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [[[.,.],[.,.]],[[.,.],[.,.]]] => [1,3,2,5,7,6,4] => [1,3,2,7,6,5,4] => 2
[1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [[[.,.],[.,.]],[[.,[.,.]],.]] => [1,3,2,6,5,7,4] => [1,3,2,7,5,6,4] => 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[[[.,.],.],.]] => [1,3,2,5,6,7,4] => [1,3,2,7,5,6,4] => 2
[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [[[.,.],[.,[.,.]]],[.,[.,.]]] => [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[[.,.],.]] => [1,4,3,2,6,7,5] => [1,4,3,2,7,6,5] => 2
[1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [[[.,.],[[.,.],.]],[.,[.,.]]] => [1,3,4,2,7,6,5] => [1,4,3,2,7,6,5] => 2
[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [[[.,.],[[.,.],.]],[[.,.],.]] => [1,3,4,2,6,7,5] => [1,4,3,2,7,6,5] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [[[.,[.,[.,[.,[.,.]]]]],.],.] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [[[.,[.,[.,[[.,.],.]]]],.],.] => [4,5,3,2,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [[[.,[.,[[.,.],[.,.]]]],.],.] => [3,5,4,2,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [[[.,[.,[[.,[.,.]],.]]],.],.] => [4,3,5,2,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [[[.,[.,[[[.,.],.],.]]],.],.] => [3,4,5,2,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [[[.,[[.,.],[.,[.,.]]]],.],.] => [2,5,4,3,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [[[.,[[.,.],[[.,.],.]]],.],.] => [2,4,5,3,1,6,7] => [5,4,3,2,1,6,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[.,.],.],.],[.,[.,[.,.]]]] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,[[.,.],.]]] => [1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[[.,.],.],.],[[.,.],[.,.]]] => [1,2,3,5,7,6,4] => [1,2,3,7,6,5,4] => 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [[[[.,.],.],.],[[.,[.,.]],.]] => [1,2,3,6,5,7,4] => [1,2,3,7,5,6,4] => 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[[.,.],.],.],[[[.,.],.],.]] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [[[[.,.],.],[.,.]],[.,[.,.]]] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[[.,.],.]] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => 2
[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],[.,.]] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [[[[.,.],.],[.,[.,[.,.]]]],.] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [[[[.,.],.],[.,[[.,.],.]]],.] => [1,2,5,6,4,3,7] => [1,2,6,5,4,3,7] => 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [[[[.,.],.],[[.,.],.]],[.,.]] => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => 2
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [[[[.,.],.],[[.,.],[.,.]]],.] => [1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [[[[.,.],.],[[.,[.,.]],.]],.] => [1,2,5,4,6,3,7] => [1,2,6,4,5,3,7] => 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [[[[.,.],.],[[[.,.],.],.]],.] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [[[[.,.],[.,.]],.],[.,[.,.]]] => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => 2
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [[[[.,.],[.,.]],.],[[.,.],.]] => [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => 2
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],[.,.]] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => 3
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [[[[.,.],[.,.]],[.,[.,.]]],.] => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [[[[.,.],[.,.]],[[.,.],.]],.] => [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => 2
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [[[[.,.],[.,[.,.]]],.],[.,.]] => [1,4,3,2,5,7,6] => [1,4,3,2,5,7,6] => 2
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [[[[.,.],[.,[.,.]]],[.,.]],.] => [1,4,3,2,6,5,7] => [1,4,3,2,6,5,7] => 2
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [[[[.,.],[[.,.],.]],.],[.,.]] => [1,3,4,2,5,7,6] => [1,4,3,2,5,7,6] => 2
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [[[[.,.],[[.,.],.]],[.,.]],.] => [1,3,4,2,6,5,7] => [1,4,3,2,6,5,7] => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,[.,.]]]],.],.],.] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [[[[.,[.,[[.,.],.]]],.],.],.] => [3,4,2,1,5,6,7] => [4,3,2,1,5,6,7] => 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [[[[.,[[.,.],[.,.]]],.],.],.] => [2,4,3,1,5,6,7] => [4,3,2,1,5,6,7] => 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[[[.,.],.],.],.],[.,[.,.]]] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[[[.,.],.],.],.],[[.,.],.]] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],[.,.]] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [[[[[.,.],.],.],[.,[.,.]]],.] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [[[[[.,.],.],.],[[.,.],.]],.] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[[[[.,.],.],[.,.]],.],[.,.]] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [[[[[.,.],.],[.,.]],[.,.]],.] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => 2
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [[[[[.,.],.],[.,[.,.]]],.],.] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [[[[[.,.],.],[[.,.],.]],.],.] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [[[[[.,.],[.,.]],.],.],[.,.]] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => 2
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [[[[[.,.],[.,.]],.],[.,.]],.] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [[[[[.,.],[.,.]],[.,.]],.],.] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [[[[[.,.],[.,[.,.]]],.],.],.] => [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [[[[[.,.],[[.,.],.]],.],.],.] => [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [[[[[.,[.,[.,.]]],.],.],.],.] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [[[[[.,[[.,.],.]],.],.],.],.] => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [[[[[[.,.],.],.],.],[.,.]],.] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [[[[[[.,.],.],.],[.,.]],.],.] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [[[[[[.,.],.],[.,.]],.],.],.] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [[[[[[.,.],[.,.]],.],.],.],.] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [[[[[[.,[.,.]],.],.],.],.],.] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]] => [6,8,7,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]] => [7,6,8,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]] => [6,7,8,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]] => [5,8,7,6,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]] => [5,7,8,6,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]] => [6,5,8,7,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]] => [7,6,5,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]] => [6,7,5,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]] => [5,6,8,7,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]] => [5,7,6,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]] => [6,5,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]] => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]] => [4,8,7,6,5,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]] => [4,7,8,6,5,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]] => [4,6,8,7,5,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]] => [4,7,6,8,5,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [.,[.,[.,[[.,.],[[[.,.],.],.]]]]] => [4,6,7,8,5,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]] => [5,4,7,8,6,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]] => [6,5,4,8,7,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [.,[.,[.,[[.,[[.,.],.]],[.,.]]]]] => [5,6,4,8,7,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]] => [4,5,8,7,6,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [.,[.,[.,[[[.,.],.],[[.,.],.]]]]] => [4,5,7,8,6,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [.,[.,[.,[[[.,.],[.,.]],[.,.]]]]] => [4,6,5,8,7,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [.,[.,[.,[[[[.,.],.],.],[.,.]]]]] => [4,5,6,8,7,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]] => [3,8,7,6,5,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]] => [3,7,8,6,5,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]] => [3,6,8,7,5,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]] => [3,7,6,8,5,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]] => [3,5,7,8,6,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [.,[.,[[.,.],[[[.,.],.],[.,.]]]]] => [3,5,6,8,7,4,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]] => [4,3,8,7,6,5,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [.,[.,[[.,[.,.]],[.,[[.,.],.]]]]] => [4,3,7,8,6,5,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]] => [6,5,4,3,8,7,2,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [.,[.,[[.,[.,[[.,.],.]]],[.,.]]]] => [5,6,4,3,8,7,2,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [.,[.,[[.,[[[.,.],.],.]],[.,.]]]] => [4,5,6,3,8,7,2,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [.,[.,[[[.,.],.],[.,[.,[.,.]]]]]] => [3,4,8,7,6,5,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]] => [3,6,5,4,8,7,2,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]] => [4,3,6,5,8,7,2,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]] => [2,8,7,6,5,4,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]] => [2,7,8,6,5,4,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [.,[[.,.],[.,[.,[[.,[.,.]],.]]]]] => [2,7,6,8,5,4,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [.,[[.,.],[.,[.,[[[.,.],.],.]]]]] => [2,6,7,8,5,4,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]] => [2,5,8,7,6,4,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]] => [2,4,8,7,6,5,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [.,[[.,.],[[.,.],[.,[[.,.],.]]]]] => [2,4,7,8,6,5,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [.,[[.,.],[[.,.],[[.,.],[.,.]]]]] => [2,4,6,8,7,5,3,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [.,[[.,.],[[.,[.,[.,.]]],[.,.]]]] => [2,6,5,4,8,7,3,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [.,[[.,.],[[.,[[.,.],.]],[.,.]]]] => [2,5,6,4,8,7,3,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [.,[[.,.],[[[.,.],[.,.]],[.,.]]]] => [2,4,6,5,8,7,3,1] => [8,7,4,3,6,5,2,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]] => [3,2,6,5,8,7,4,1] => [8,2,7,4,6,5,3,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]] => [4,3,2,8,7,6,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [.,[[.,[.,[.,.]]],[.,[[.,.],.]]]] => [4,3,2,7,8,6,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [.,[[.,[.,[.,.]]],[[.,.],[.,.]]]] => [4,3,2,6,8,7,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]] => [6,5,4,3,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]] => [4,6,5,3,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]] => [5,4,6,3,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]] => [4,5,6,3,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [.,[[.,[[.,.],.]],[.,[.,[.,.]]]]] => [3,4,2,8,7,6,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [.,[[.,[[.,.],.]],[.,[[.,.],.]]]] => [3,4,2,7,8,6,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [.,[[.,[[.,.],[.,[.,.]]]],[.,.]]] => [3,6,5,4,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]] => [3,5,6,4,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]] => [4,3,6,5,2,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [.,[[[.,.],.],[[.,[.,.]],[.,.]]]] => [2,3,6,5,8,7,4,1] => [8,2,7,4,6,5,3,1] => 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => [8,2,7,4,6,5,3,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [.,[[[.,.],[.,.]],[.,[.,[.,.]]]]] => [2,4,3,8,7,6,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]] => [2,4,3,6,8,7,5,1] => [8,3,2,7,6,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [.,[[[.,.],[.,[.,[.,.]]]],[.,.]]] => [2,6,5,4,3,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0] => [.,[[[.,.],[[.,.],[.,.]]],[.,.]]] => [2,4,6,5,3,8,7,1] => [8,5,4,3,2,7,6,1] => 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [.,[[[.,[.,[.,.]]],[.,.]],[.,.]]] => [4,3,2,6,5,8,7,1] => [8,3,2,5,4,7,6,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [.,[[[.,[[.,.],.]],[.,.]],[.,.]]] => [3,4,2,6,5,8,7,1] => [8,3,2,5,4,7,6,1] => 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [.,[[[[.,.],[.,.]],[.,.]],[.,.]]] => [2,4,3,6,5,8,7,1] => [8,3,2,5,4,7,6,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]] => [1,7,6,8,5,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]] => [1,6,7,8,5,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]] => [1,6,5,8,7,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]] => [1,7,6,5,8,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0] => [[.,.],[.,[.,[[.,[[.,.],.]],.]]]] => [1,6,7,5,8,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0] => [[.,.],[.,[.,[[[.,.],[.,.]],.]]]] => [1,5,7,6,8,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0] => [[.,.],[.,[.,[[[.,[.,.]],.],.]]]] => [1,6,5,7,8,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]] => [1,5,6,7,8,4,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]] => [1,4,8,7,6,5,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]] => [1,4,7,8,6,5,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[.,.],[.,[[.,.],[[.,.],[.,.]]]]] => [1,4,6,8,7,5,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[.,.],[.,[[.,.],[[[.,.],.],.]]]] => [1,4,6,7,8,5,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[.,.],[.,[[.,[.,.]],[[.,.],.]]]] => [1,5,4,7,8,6,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0] => [[.,.],[.,[[.,[.,[.,.]]],[.,.]]]] => [1,6,5,4,8,7,3,2] => [1,8,7,4,6,5,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]] => [1,7,6,5,4,8,3,2] => [1,8,7,5,4,6,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[.,.],[.,[[[.,.],.],[.,[.,.]]]]] => [1,4,5,8,7,6,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[.,.],[.,[[[.,.],.],[[.,.],.]]]] => [1,4,5,7,8,6,3,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0] => [[.,.],[.,[[[.,.],[.,.]],[.,.]]]] => [1,4,6,5,8,7,3,2] => [1,8,7,4,6,5,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[.,.],[.,[[[.,.],[.,[.,.]]],.]]] => [1,4,7,6,5,8,3,2] => [1,8,7,5,4,6,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0] => [[.,.],[.,[[[.,.],[[.,.],.]],.]]] => [1,4,6,7,5,8,3,2] => [1,8,7,5,4,6,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0] => [[.,.],[.,[[[.,[.,.]],.],[.,.]]]] => [1,5,4,6,8,7,3,2] => [1,8,7,4,6,5,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0] => [[.,.],[.,[[[.,[.,[.,.]]],.],.]]] => [1,6,5,4,7,8,3,2] => [1,8,7,4,5,6,3,2] => 1
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0] => [[.,.],[.,[[[.,[[.,.],.]],.],.]]] => [1,5,6,4,7,8,3,2] => [1,8,7,4,5,6,3,2] => 1
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0] => [[.,.],[.,[[[[.,[.,.]],.],.],.]]] => [1,5,4,6,7,8,3,2] => [1,8,7,4,5,6,3,2] => 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[.,.],[.,[[[[[.,.],.],.],.],.]]] => [1,4,5,6,7,8,3,2] => [1,8,7,4,5,6,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]] => [1,3,8,7,6,5,4,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]] => [1,3,7,8,6,5,4,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]] => [1,3,6,8,7,5,4,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[.,.],[[.,.],[.,[[[.,.],.],.]]]] => [1,3,6,7,8,5,4,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]] => [1,3,5,8,7,6,4,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => [1,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0] => [[.,.],[[.,.],[[.,[[.,.],.]],.]]] => [1,3,6,7,5,8,4,2] => [1,8,7,5,4,6,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]] => [1,3,5,6,8,7,4,2] => [1,8,7,4,6,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]] => [1,3,5,7,6,8,4,2] => [1,8,7,5,4,6,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0] => [[.,.],[[.,.],[[[.,[.,.]],.],.]]] => [1,3,6,5,7,8,4,2] => [1,8,7,4,5,6,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[.,.],[[.,.],[[[[.,.],.],.],.]]] => [1,3,5,6,7,8,4,2] => [1,8,7,4,5,6,3,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]] => [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]] => [2,1,7,8,6,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[.,[.,.]],[.,[.,[[.,.],[.,.]]]]] => [2,1,6,8,7,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[.,[.,.]],[.,[.,[[.,[.,.]],.]]]] => [2,1,7,6,8,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,[[[.,.],.],.]]]] => [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]] => [2,1,5,8,7,6,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]] => [2,1,5,7,8,6,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]] => [2,1,6,5,8,7,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[.,[.,.]],[.,[[[.,.],.],[.,.]]]] => [2,1,5,6,8,7,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]] => [2,1,4,8,7,6,5,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]] => [2,1,4,7,8,6,5,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]] => [2,1,4,6,8,7,5,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]] => [2,1,6,5,4,8,7,3] => [2,1,8,5,4,7,6,3] => 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0] => [[.,[.,.]],[[.,[[.,.],.]],[.,.]]] => [2,1,5,6,4,8,7,3] => [2,1,8,5,4,7,6,3] => 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0] => [[.,[.,.]],[[[.,.],[.,.]],[.,.]]] => [2,1,4,6,5,8,7,3] => [2,1,8,5,4,7,6,3] => 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]] => [3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]] => [3,2,1,7,8,6,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]] => [3,2,1,6,8,7,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[.,[.,[.,.]]],[.,[[.,[.,.]],.]]] => [3,2,1,7,6,8,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0] => [[.,[.,[.,.]]],[.,[[[.,.],.],.]]] => [3,2,1,6,7,8,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]] => [3,2,1,5,8,7,6,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]] => [3,2,1,5,7,8,6,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]] => [4,3,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]] => [4,3,2,1,6,8,7,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]] => [5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]] => [5,4,3,2,1,7,8,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.] => [6,7,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0] => [[.,[.,[.,[.,[[.,.],.]]]]],[.,.]] => [5,6,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0] => [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.] => [5,7,6,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.] => [6,5,7,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.] => [5,6,7,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0] => [[.,[.,[.,[[.,.],.]]]],[.,[.,.]]] => [4,5,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0] => [[.,[.,[.,[[.,.],[.,.]]]]],[.,.]] => [4,6,5,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0] => [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.] => [4,7,6,5,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0] => [[.,[.,[.,[[.,.],[[.,.],.]]]]],.] => [4,6,7,5,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0] => [[.,[.,[.,[[.,[.,.]],.]]]],[.,.]] => [5,4,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.] => [5,4,7,6,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.] => [6,5,4,7,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.] => [5,6,4,7,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[.,[.,[.,[[[.,.],.],.]]]],[.,.]] => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0] => [[.,[.,[.,[[[.,.],.],[.,.]]]]],.] => [4,5,7,6,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0] => [[.,[.,[.,[[[.,.],[.,.]],.]]]],.] => [4,6,5,7,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[.,[.,[.,[[[.,[.,.]],.],.]]]],.] => [5,4,6,7,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.] => [4,5,6,7,3,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0] => [[.,[.,[[.,.],.]]],[.,[.,[.,.]]]] => [3,4,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0] => [[.,[.,[[.,.],.]]],[.,[[.,.],.]]] => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0] => [[.,[.,[[.,.],.]]],[[.,.],[.,.]]] => [3,4,2,1,6,8,7,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0] => [[.,[.,[[.,.],[.,.]]]],[.,[.,.]]] => [3,5,4,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0] => [[.,[.,[[.,.],[.,.]]]],[[.,.],.]] => [3,5,4,2,1,7,8,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0] => [[.,[.,[[.,.],[.,[.,.]]]]],[.,.]] => [3,6,5,4,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0] => [[.,[.,[[.,.],[[.,.],.]]]],[.,.]] => [3,5,6,4,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0] => [[.,[.,[[.,.],[[.,.],[.,.]]]]],.] => [3,5,7,6,4,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0] => [[.,[.,[[.,.],[[[.,.],.],.]]]],.] => [3,5,6,7,4,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0] => [[.,[.,[[.,[.,.]],.]]],[.,[.,.]]] => [4,3,5,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0] => [[.,[.,[[.,[.,.]],[.,.]]]],[.,.]] => [4,3,6,5,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0] => [[.,[.,[[.,[.,.]],[.,[.,.]]]]],.] => [4,3,7,6,5,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0] => [[.,[.,[[.,[.,[.,.]]],[.,.]]]],.] => [5,4,3,7,6,2,1,8] => [7,6,3,5,4,2,1,8] => 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.] => [6,5,4,3,7,2,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.] => [5,6,4,3,7,2,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0] => [[.,[.,[[.,[[.,.],.]],[.,.]]]],.] => [4,5,3,7,6,2,1,8] => [7,6,3,5,4,2,1,8] => 1
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.] => [5,4,6,3,7,2,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0] => [[.,[.,[[[.,.],.],.]]],[.,[.,.]]] => [3,4,5,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0] => [[.,[.,[[[.,.],.],.]]],[[.,.],.]] => [3,4,5,2,1,7,8,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0] => [[.,[.,[[[.,.],.],[.,.]]]],[.,.]] => [3,4,6,5,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0] => [[.,[.,[[[.,.],.],[.,[.,.]]]]],.] => [3,4,7,6,5,2,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0] => [[.,[.,[[[.,.],[.,.]],[.,.]]]],.] => [3,5,4,7,6,2,1,8] => [7,6,3,5,4,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0] => [[.,[.,[[[.,.],[.,[.,.]]],.]]],.] => [3,6,5,4,7,2,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0] => [[.,[.,[[[.,.],[[.,.],.]],.]]],.] => [3,5,6,4,7,2,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0] => [[.,[.,[[[.,[.,.]],.],[.,.]]]],.] => [4,3,5,7,6,2,1,8] => [7,6,3,5,4,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0] => [[.,[.,[[[.,[.,.]],[.,.]],.]]],.] => [4,3,6,5,7,2,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.] => [5,4,3,6,7,2,1,8] => [7,6,3,4,5,2,1,8] => 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0] => [[.,[.,[[[[[.,.],.],.],.],.]]],.] => [3,4,5,6,7,2,1,8] => [7,6,3,4,5,2,1,8] => 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]] => [2,3,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[.,[[.,.],.]],[.,[.,[[.,.],.]]]] => [2,3,1,7,8,6,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[.,[[.,.],.]],[.,[[.,.],[.,.]]]] => [2,3,1,6,8,7,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0] => [[.,[[.,.],.]],[.,[[.,[.,.]],.]]] => [2,3,1,7,6,8,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[.,[[.,.],.]],[.,[[[.,.],.],.]]] => [2,3,1,6,7,8,5,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[.,[[.,.],.]],[[.,.],[[.,.],.]]] => [2,3,1,5,7,8,6,4] => [3,2,1,8,7,6,5,4] => 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]] => [2,4,3,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0] => [[.,[[.,.],[.,.]]],[.,[[.,.],.]]] => [2,4,3,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0] => [[.,[[.,.],[.,.]]],[[.,.],[.,.]]] => [2,4,3,1,6,8,7,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0] => [[.,[[.,.],[.,[.,.]]]],[.,[.,.]]] => [2,5,4,3,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0] => [[.,[[.,.],[.,[.,.]]]],[[.,.],.]] => [2,5,4,3,1,7,8,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0] => [[.,[[.,.],[.,[.,[.,.]]]]],[.,.]] => [2,6,5,4,3,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[.,[[.,.],[.,[.,[.,[.,.]]]]]],.] => [2,7,6,5,4,3,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0] => [[.,[[.,.],[.,[.,[[.,.],.]]]]],.] => [2,6,7,5,4,3,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0] => [[.,[[.,.],[.,[[.,.],.]]]],[.,.]] => [2,5,6,4,3,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0] => [[.,[[.,.],[.,[[.,.],[.,.]]]]],.] => [2,5,7,6,4,3,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0] => [[.,[[.,.],[.,[[[.,.],.],.]]]],.] => [2,5,6,7,4,3,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0] => [[.,[[.,.],[[.,.],.]]],[.,[.,.]]] => [2,4,5,3,1,8,7,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0] => [[.,[[.,.],[[.,.],.]]],[[.,.],.]] => [2,4,5,3,1,7,8,6] => [5,4,3,2,1,8,7,6] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0] => [[.,[[.,.],[[.,.],[.,.]]]],[.,.]] => [2,4,6,5,3,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0] => [[.,[[.,.],[[.,.],[[.,.],.]]]],.] => [2,4,6,7,5,3,1,8] => [7,6,5,4,3,2,1,8] => 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0] => [[.,[[.,.],[[.,[.,.]],[.,.]]]],.] => [2,5,4,7,6,3,1,8] => [7,6,3,5,4,2,1,8] => 1
[1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0] => [[.,[[.,.],[[.,[[.,.],.]],.]]],.] => [2,5,6,4,7,3,1,8] => [7,6,4,3,5,2,1,8] => 1
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0] => [[.,[[.,.],[[[.,[.,.]],.],.]]],.] => [2,5,4,6,7,3,1,8] => [7,6,3,4,5,2,1,8] => 1
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0] => [[.,[[.,.],[[[[.,.],.],.],.]]],.] => [2,4,5,6,7,3,1,8] => [7,6,3,4,5,2,1,8] => 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0] => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]] => [3,2,4,1,7,6,8,5] => [4,2,3,1,8,6,7,5] => 2
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0] => [[.,[[.,[.,.]],.]],[[[.,.],.],.]] => [3,2,4,1,6,7,8,5] => [4,2,3,1,8,6,7,5] => 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0] => [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]] => [4,3,2,6,5,1,8,7] => [6,3,2,5,4,1,8,7] => 2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0] => [[.,[[.,[[.,.],.]],[.,.]]],[.,.]] => [3,4,2,6,5,1,8,7] => [6,3,2,5,4,1,8,7] => 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,1,0,0] => [[.,[[[.,.],.],.]],[[.,[.,.]],.]] => [2,3,4,1,7,6,8,5] => [4,2,3,1,8,6,7,5] => 2
[1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[.,[[[.,.],.],.]],[[[.,.],.],.]] => [2,3,4,1,6,7,8,5] => [4,2,3,1,8,6,7,5] => 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0] => [[.,[[[.,.],[.,.]],[.,.]]],[.,.]] => [2,4,3,6,5,1,8,7] => [6,3,2,5,4,1,8,7] => 2
[1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 3
[1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],[.,.]],[.,[[.,.],.]]] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 3
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],[[.,.],[.,.]]] => [2,1,4,3,6,8,7,5] => [2,1,4,3,8,7,6,5] => 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]] => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0] => [[[.,[.,.]],[.,[[.,.],.]]],[.,.]] => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0] => [[[.,[.,.]],[[.,.],[.,.]]],[.,.]] => [2,1,4,6,5,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.] => [5,6,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.] => [5,4,6,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [[[.,[.,[.,[[[.,.],.],.]]]],.],.] => [4,5,6,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0] => [[[.,[.,[[.,.],.]]],[.,.]],[.,.]] => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0] => [[[.,[.,[[.,.],[.,[.,.]]]]],.],.] => [3,6,5,4,2,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0] => [[[.,[.,[[.,[.,.]],[.,.]]]],.],.] => [4,3,6,5,2,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0] => [[[.,[[.,.],[.,.]]],[.,.]],[.,.]] => [2,4,3,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0] => [[[.,[[.,.],[[.,.],[.,.]]]],.],.] => [2,4,6,5,3,1,7,8] => [6,5,4,3,2,1,7,8] => 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.] => [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5,8] => 2
[1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0] => [[[[.,.],[.,[.,.]]],[[.,.],.]],.] => [1,4,3,2,6,7,5,8] => [1,4,3,2,7,6,5,8] => 2
[1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0] => [[[[.,.],[[.,.],.]],[[.,.],.]],.] => [1,3,4,2,6,7,5,8] => [1,4,3,2,7,6,5,8] => 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]] => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0] => [[[[.,[.,[.,.]]],.],.],[[.,.],.]] => [3,2,1,4,5,7,8,6] => [3,2,1,4,5,8,7,6] => 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.] => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.] => [4,5,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0] => [[[[.,[.,[[.,.],[.,.]]]],.],.],.] => [3,5,4,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0] => [[[[.,[.,[[.,[.,.]],.]]],.],.],.] => [4,3,5,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [[[[.,[.,[[[.,.],.],.]]],.],.],.] => [3,4,5,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0] => [[[[.,[[.,.],.]],.],.],[[.,.],.]] => [2,3,1,4,5,7,8,6] => [3,2,1,4,5,8,7,6] => 2
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [[[[.,[[.,.],[.,[.,.]]]],.],.],.] => [2,5,4,3,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0] => [[[[.,[[.,.],[[.,.],.]]],.],.],.] => [2,4,5,3,1,6,7,8] => [5,4,3,2,1,6,7,8] => 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8] => 3
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [[[[[.,[.,[[.,.],.]]],.],.],.],.] => [3,4,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [[[[[.,[[.,.],[.,.]]],.],.],.],.] => [2,4,3,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0] => [[[[[[.,.],.],.],[.,.]],[.,.]],.] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0] => [[[[[[.,.],.],[.,.]],[.,.]],.],.] => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [[[[[[.,.],[.,.]],.],.],[.,.]],.] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => 2
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [[[[[[.,.],[.,.]],[.,.]],.],.],.] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [[[[[[.,[.,.]],.],.],.],.],[.,.]] => [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [[[[[[.,[.,[.,.]]],.],.],.],.],.] => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [[[[[[.,[[.,.],.]],.],.],.],.],.] => [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [[[[[[[.,.],.],.],.],.],[.,.]],.] => [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [[[[[[[.,.],.],.],[.,.]],.],.],.] => [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [[[[[[[.,.],[.,.]],.],.],.],.],.] => [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [[[[[[[.,[.,.]],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[[[[[[[.,.],.],.],.],.],.],.],.] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.] => [7,8,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.] => [7,6,8,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],.] => [6,7,8,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]],.] => [6,5,8,7,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]],.] => [7,6,5,8,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]],.] => [6,7,5,8,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]],.] => [6,5,7,8,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0] => [[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]],.] => [6,5,4,7,8,3,2,1,9] => [8,7,6,4,5,3,2,1,9] => 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]],.] => [2,8,7,6,5,4,3,1,9] => [8,7,6,5,4,3,2,1,9] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.] => [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.] => [6,7,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.] => [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0] => [[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.] => [5,6,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.] => [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 1
[1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [[[[[.,[.,[.,[[.,.],.]]]],.],.],.],.] => [4,5,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.] => [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [[[[[[.,[.,[[.,.],.]]],.],.],.],.],.] => [3,4,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [[[[[[.,[[.,.],[.,.]]],.],.],.],.],.] => [2,4,3,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [[[[[[[.,[.,[.,.]]],.],.],.],.],.],.] => [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [[[[[[[.,[[.,.],.]],.],.],.],.],.],.] => [2,3,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] => [8,9,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]] => [8,7,9,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]] => [8,7,6,9,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]] => [8,7,6,5,9,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]] => [9,10,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]] => [7,8,9,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]] => [9,8,10,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]] => [6,8,7,9,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]] => [9,8,7,10,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [.,[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]] => [5,8,7,6,9,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]] => [9,8,7,6,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.] => [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]] => [8,9,10,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]] => [6,7,8,9,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.] => [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]] => [7,8,9,10,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]] => [5,6,7,8,9,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]] => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]] => [7,9,8,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]] => [4,9,8,7,6,5,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]] => [3,9,8,7,6,5,4,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]] => [2,9,8,7,6,5,4,3,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]] => [2,8,9,7,6,5,4,3,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [1,8,9,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]] => [6,10,9,8,7,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] => [1,9,10,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]] => [1,3,5,7,9,10,8,6,4,2] => [1,10,9,8,7,6,5,4,3,2] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]] => [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]] => [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]] => [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 2
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.],.] => [5,6,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0] => [[[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.],.] => [6,7,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [1,11,10,9,8,7,6,5,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]] => [1,8,7,9,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.] => [8,9,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.] => [2,3,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.],.] => [3,4,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[.,.],[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]] => [1,9,8,10,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [.,[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]] => [2,8,7,9,6,5,4,3,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.] => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]],.] => [7,8,9,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]],.] => [7,6,8,9,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0] => [[.,[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]],.] => [7,6,5,8,9,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.] => [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.] => [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9,10] => 1
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.] => [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.] => [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.] => [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.] => [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]] => [8,7,6,5,4,3,2,10,9,1] => [10,7,6,5,4,3,2,9,8,1] => 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => [.,[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]]] => [2,4,3,6,5,8,7,10,9,1] => [10,3,2,5,4,7,6,9,8,1] => 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => [.,[[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]] => [2,4,3,6,5,8,7,10,9,12,11,1] => [12,3,2,5,4,7,6,9,8,11,10,1] => 1
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => [[[[.,[[.,.],[.,.]]],[.,.]],[.,.]],[.,.]] => [2,4,3,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Generating function
click to show known generating functions
Search the OEIS for these generating functions
Search the Online Encyclopedia of Integer
Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 1,4 1,12,1 1,34,7 1,98,32,1
F1=1
F2=1+q
F3=1+4 q
F4=1+12 q+q2
F5=1+34 q+7 q2
F6=1+98 q+32 q2+q3
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation π=[π1,…,πn], this statistic counts the number of position j such that πj<j and there do not exist indices i,k with i<j<k and πi>πj>πk.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation π=[π1,…,πn], this statistic counts the number of position j such that πj<j and there do not exist indices i,k with i<j<k and πi>πj>πk.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
Map
to binary tree: up step, left tree, down step, right tree
Description
Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path D of semilength n with n>1 may be uniquely decomposed into 1L0R for Dyck paths L,R of respective semilengths n1,n2 with n1+n2=n−1.
This map sends D to the binary tree T consisting of a root node with a left child according to L and a right child according to R and then recursively proceeds.
The base case of the unique Dyck path of semilength 1 is sent to a single node.
A Dyck path D of semilength n with n>1 may be uniquely decomposed into 1L0R for Dyck paths L,R of respective semilengths n1,n2 with n1+n2=n−1.
This map sends D to the binary tree T consisting of a root node with a left child according to L and a right child according to R and then recursively proceeds.
The base case of the unique Dyck path of semilength 1 is sent to a single node.
Map
Demazure product with inverse
Description
This map sends a permutation π to π−1⋆π where ⋆ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations π for which π=π−1.
This map is a surjection onto the set of involutions, i.e., the set of permutations π for which π=π−1.
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.
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.
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