- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000215: Permutations ⟶ ℤ

Values

[1,0] => [1] => [.,.] => [1] => 1

[1,0,1,0] => [1,2] => [.,[.,.]] => [2,1] => 2

[1,1,0,0] => [2,1] => [[.,.],.] => [1,2] => 0

[1,0,1,0,1,0] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 3

[1,0,1,1,0,0] => [1,3,2] => [.,[[.,.],.]] => [2,3,1] => 1

[1,1,0,0,1,0] => [2,1,3] => [[.,.],[.,.]] => [1,3,2] => 1

[1,1,0,1,0,0] => [2,3,1] => [[.,.],[.,.]] => [1,3,2] => 1

[1,1,1,0,0,0] => [3,2,1] => [[[.,.],.],.] => [1,2,3] => 0

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 4

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 2

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 2

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 2

[1,0,1,1,1,0,0,0] => [1,4,3,2] => [.,[[[.,.],.],.]] => [2,3,4,1] => 1

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 2

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [[.,.],[[.,.],.]] => [1,3,4,2] => 0

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 2

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 2

[1,1,0,1,1,0,0,0] => [2,4,3,1] => [[.,.],[[.,.],.]] => [1,3,4,2] => 0

[1,1,1,0,0,0,1,0] => [3,2,1,4] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1

[1,1,1,0,0,1,0,0] => [3,2,4,1] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1

[1,1,1,0,1,0,0,0] => [3,4,2,1] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1

[1,1,1,1,0,0,0,0] => [4,3,2,1] => [[[[.,.],.],.],.] => [1,2,3,4] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 5

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 3

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 3

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 3

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 2

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 3

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 1

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 3

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 3

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 1

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 2

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 2

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 2

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 3

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 0

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 3

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 3

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 3

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 0

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 0

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 0

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 0

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 4

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 4

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 4

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 3

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 4

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 1

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 1

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 1

>>> Load all 417 entries. <<<

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2

[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2

[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2

[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 1

[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 4

[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 2

[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 1

[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0

[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0

[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => 0

[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 4

[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 2

[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 1

[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 2

[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 2

[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2

[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 1

[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0

[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0

[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2

[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,5,3,1] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0

[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 1

[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => 0

[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1

[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => [[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 0

[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1

[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1

[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,6,4,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 0

[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,2,1,6,5] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1

[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,2,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1

[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => [[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3

[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,5,2,1] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1

[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,4,2,1,6] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,4,6,2,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 1

[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 0

[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 0

[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 0

[1,1,1,1,0,0,1,0,0,0,1,0] => [4,3,5,2,1,6] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,0,1,0,0,1,0,0] => [4,3,5,2,6,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,5,6,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,0,1,1,0,0,0,0] => [4,3,6,5,2,1] => [[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 0

[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,3,2,1,6] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,3,6,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2

[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,5,3,2,1] => [[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 0

[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 1

[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 1

[1,1,1,1,1,0,0,0,1,0,0,0] => [5,4,3,6,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 1

[1,1,1,1,1,0,0,1,0,0,0,0] => [5,4,6,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 1

[1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,4,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 1

[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 0

[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => 7

[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]] => [2,3,4,5,6,7,1] => 1

[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 5

[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => 3

[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,1,3,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,1,3,4,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,1,3,4,7,6,5] => [[.,.],[.,[.,[[[.,.],.],.]]]] => [1,5,6,7,4,3,2] => 2

[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [2,1,3,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [2,1,3,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [2,1,3,5,7,6,4] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,5,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [2,1,4,5,3,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [2,1,4,5,6,3,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [2,1,4,5,6,7,3] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,1,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 5

[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [2,3,1,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => 3

[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [2,3,1,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [2,3,1,4,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [2,3,1,4,7,6,5] => [[.,.],[.,[.,[[[.,.],.],.]]]] => [1,5,6,7,4,3,2] => 2

[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [2,3,1,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [2,3,1,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [2,3,1,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [2,3,1,5,7,6,4] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [2,3,4,1,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 5

[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [2,3,4,1,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => 3

[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [2,3,4,1,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [2,3,4,1,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [2,3,4,1,7,6,5] => [[.,.],[.,[.,[[[.,.],.],.]]]] => [1,5,6,7,4,3,2] => 2

[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,5,1,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 5

[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,1,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => 3

[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,1,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 5

[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 5

[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,7,6,1] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => 3

[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [2,3,4,6,5,1,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [2,3,4,6,5,7,1] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [2,3,4,6,7,5,1] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 3

[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [2,3,4,7,6,5,1] => [[.,.],[.,[.,[[[.,.],.],.]]]] => [1,5,6,7,4,3,2] => 2

[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [2,3,5,4,1,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [2,3,5,4,1,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [2,3,5,4,6,1,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [2,3,5,4,6,7,1] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [2,3,5,4,7,6,1] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [2,3,5,6,4,1,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [2,3,5,6,4,7,1] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [2,3,5,6,7,4,1] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 3

[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [2,3,5,7,6,4,1] => [[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 1

[1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [2,4,3,1,5,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [2,4,3,5,1,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [2,4,3,5,6,1,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [2,4,3,5,6,7,1] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [2,4,5,3,1,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [2,4,5,3,6,1,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [2,4,5,3,6,7,1] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,1,0,1,0,0,0,1,0] => [2,4,5,6,3,1,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [2,4,5,6,3,7,1] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [2,4,5,6,7,3,1] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 3

[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 1

[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,5,4,3,2,7,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 1

[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [6,5,4,3,7,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 1

[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [6,5,4,7,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 1

[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [6,5,7,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 1

[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [6,7,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 1

[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 0

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [8,7,6,5,4,3,2,1] => 8

[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,2,4,3,6,7,5,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [1,2,4,3,6,7,8,5] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,2,4,6,5,3,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,2,4,6,5,7,3,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [1,2,4,6,7,5,3,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [1,2,4,6,7,8,5,3] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => [3,5,8,7,6,4,2,1] => 4

[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,3,2,4,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [1,3,2,4,5,7,8,6] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [1,3,4,2,5,7,8,6] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [1,3,4,5,2,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,5,2,7,8,6] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [1,3,4,5,7,6,2,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,5,7,6,8,2] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [1,3,4,5,7,8,6,2] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => [2,6,8,7,5,4,3,1] => 4

[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [1,4,3,2,7,6,8,5] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [1,4,3,7,6,8,5,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [1,4,3,7,8,6,5,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,4,7,6,5,3,2,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [1,4,7,6,5,8,3,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [1,4,7,8,6,5,3,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]] => [2,3,5,6,8,7,4,1] => 2

[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]] => [2,3,4,5,6,7,8,1] => 1

[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,6,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => 4

[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,3,4,5,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [2,1,3,4,5,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,1,3,4,5,8,7,6] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]] => [1,6,7,8,5,4,3,2] => 3

[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,3,4,6,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0] => [2,1,3,4,6,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,3,6,8,7,5] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0] => [2,1,4,6,5,3,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0] => [2,1,4,6,5,8,7,3] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0] => [2,1,4,6,8,7,5,3] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,1,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,3,1,4,5,6,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => 4

[1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,3,1,4,5,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,3,1,4,5,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,3,1,4,6,7,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0] => [2,3,1,4,6,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,4,1,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5,6,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0] => [2,3,4,1,5,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0] => [2,3,4,1,5,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,8,7,6] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]] => [1,6,7,8,5,4,3,2] => 3

[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0] => [2,3,4,1,6,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0] => [2,3,4,1,6,7,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0] => [2,3,4,1,6,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [2,3,4,5,1,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0] => [2,3,4,5,1,6,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0] => [2,3,4,5,1,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0] => [2,3,4,5,1,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0] => [2,3,4,5,1,8,7,6] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]] => [1,6,7,8,5,4,3,2] => 3

[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,5,6,1,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,6,1,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,7,1,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,8,7,6,5,4,3,2] => 6

[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,6,8,7,1] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => [1,7,8,6,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0] => [2,3,4,5,7,6,1,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0] => [2,3,4,5,7,6,8,1] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [2,3,4,5,7,8,6,1] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => [1,6,8,7,5,4,3,2] => 4

[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [2,3,4,5,8,7,6,1] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]] => [1,6,7,8,5,4,3,2] => 3

[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0] => [2,3,4,6,5,1,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0] => [2,3,4,6,5,7,1,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0] => [2,3,4,6,5,7,8,1] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0] => [2,3,4,6,7,5,1,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0] => [2,3,4,6,7,5,8,1] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [2,3,4,6,7,8,5,1] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => [1,5,8,7,6,4,3,2] => 4

[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0] => [2,4,3,1,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0] => [2,4,3,1,6,8,7,5] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0] => [2,4,3,6,5,1,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,8,7,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0] => [2,4,3,6,8,7,5,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0] => [2,4,6,5,3,1,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0] => [2,4,6,5,3,8,7,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0] => [2,4,6,5,8,7,3,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,4,6,8,7,5,3,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 0

[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0] => [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]] => [1,2,6,7,8,5,4,3] => 2

[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0] => [3,2,4,1,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]] => [1,2,6,7,8,5,4,3] => 2

[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0] => [3,2,4,5,1,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]] => [1,2,6,7,8,5,4,3] => 2

[1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0] => [3,4,2,5,1,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]] => [1,2,6,7,8,5,4,3] => 2

[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0] => [3,4,2,5,8,7,6,1] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]] => [1,2,6,7,8,5,4,3] => 2

[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0] => [3,4,5,2,1,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]] => [1,2,6,7,8,5,4,3] => 2

[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]] => [1,2,3,5,6,7,8,4] => 0

[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0] => [4,3,2,8,7,6,5,1] => [[[[.,.],.],.],[[[[.,.],.],.],.]] => [1,2,3,5,6,7,8,4] => 0

[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0] => [4,3,8,7,6,5,2,1] => [[[[.,.],.],.],[[[[.,.],.],.],.]] => [1,2,3,5,6,7,8,4] => 0

[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [4,8,7,6,5,3,2,1] => [[[[.,.],.],.],[[[[.,.],.],.],.]] => [1,2,3,5,6,7,8,4] => 0

[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [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] => [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => 1

[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => 1

[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => 1

[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => 1

[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => 1

[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,8,7] => 1

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.] => [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] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,9,8,7,6,5,4,3,2] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]] => [2,3,4,5,6,7,8,9,1] => 1

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [9,8,7,6,5,4,3,2,1,10] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,10,9,8,7,6,5,4,3,2] => [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]] => [2,3,4,5,6,7,8,9,10,1] => 1

[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,9,11,10] => 1

[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,11,10,9,8,7,6,5,4,3,2] => [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]] => [2,3,4,5,6,7,8,9,10,11,1] => 1

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,6,7,9,8,1] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [1,8,9,7,6,5,4,3,2] => 5

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [8,7,6,5,4,3,2,9,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [8,7,6,5,4,3,9,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [8,7,6,5,4,9,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [8,7,6,5,9,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [8,7,6,9,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [8,7,9,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,9,8] => 1

[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [8,9,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]] => [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] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.] => [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] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [9,8,7,6,5,4,3,2,1] => 9

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [10,9,8,7,6,5,4,3,2,1] => 10

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [1,11,10,9,8,7,6,5,4,3,2] => 9

[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [1,8,9,7,6,5,4,3,2] => 5

[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,6,7,8,10,9] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] => [1,9,10,8,7,6,5,4,3,2] => 6

[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [10,9,8,7,6,5,4,3,2,1] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.] => [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] => [2,3,1,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,1,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [2,3,4,5,6,1,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,5,6,7,1,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,7,8,1,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => [1,9,8,7,6,5,4,3,2] => 7

[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,1,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,1,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,5,6,7,8,1,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,7,8,9,1,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] => [1,10,9,8,7,6,5,4,3,2] => 8

[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7,8,9,10,11] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [1,11,10,9,8,7,6,5,4,3,2] => 9

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,6,7,8,10,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] => [1,9,10,8,7,6,5,4,3,2] => 6

[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [9,8,10,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [10,9,11,8,7,6,5,4,3,2,1] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,9,11,10] => 1

[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,1,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [1,8,9,7,6,5,4,3,2] => 5

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0] => [9,8,7,6,5,4,3,2,10,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0] => [10,9,8,7,6,5,4,3,2,11,1] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,9,11,10] => 1

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [9,10,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [10,11,9,8,7,6,5,4,3,2,1] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,9,11,10] => 1

[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [1,8,9,7,6,5,4,3,2] => 5

[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,6,7,1,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => [1,8,9,7,6,5,4,3,2] => 5

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0] => [9,8,7,6,5,4,3,10,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0] => [9,8,7,6,5,4,10,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0] => [9,8,7,6,5,10,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [9,8,7,6,10,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [9,8,7,10,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] => [1,2,3,4,5,6,7,8,10,9] => 1

[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,1,4,5,6,7,8,9,10,11] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [1,11,10,9,8,7,6,5,4,3,2] => 9

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,7,8,9,10,1,11] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => [1,11,10,9,8,7,6,5,4,3,2] => 9

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,6,7,8,1,10,9] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] => [1,9,10,8,7,6,5,4,3,2] => 6

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$F_{1} = q$

$F_{2} = 1 + q^{2}$

$F_{3} = 1 + 3\ q + q^{3}$

$F_{4} = 3 + 4\ q + 6\ q^{2} + q^{4}$

$F_{5} = 6 + 15\ q + 10\ q^{2} + 10\ q^{3} + q^{5}$

$F_{6} = 15 + 36\ q + 45\ q^{2} + 20\ q^{3} + 15\ q^{4} + q^{6}$

Description

The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form

$(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic,

$\operatorname{adj_0}$ , counts adjacencies in the word with a zero appended.

$(\operatorname{adj_0}, \operatorname{des})$ and

$(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

Map

binary search tree: left to right

Description

Return the shape of the binary search tree of the permutation as a non labelled binary tree.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

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.