- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000425: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 4\ q + q^{3}$

$F_{5} = 21 + 15\ q + 5\ q^{3} + q^{6}$

$F_{6} = 51 + 50\ q + 3\ q^{2} + 21\ q^{3} + 6\ q^{6} + q^{10}$

Description

The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.

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 321-avoiding permutation

Description

Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.

Map

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.