- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001594: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 1 + q$

$F_{3} = 3 + 2\ q$

$F_{4} = 9 + 5\ q$

$F_{5} = 28 + 14\ q$

$F_{6} = 90 + 42\ q$

Description

The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied.
See the link for the definition.

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 Dyck path: up step, left tree, down step, right tree

Description

Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:

a leaf is associated to the empty Dyck Word
a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$ .