Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000061: Binary trees ⟶ ℤ
Values
[1,0,1,0] => [2,1] => [2,1] => [[.,.],.] => 2
[1,1,0,0] => [1,2] => [1,2] => [.,[.,.]] => 1
[1,0,1,0,1,0] => [3,2,1] => [2,3,1] => [[.,.],[.,.]] => 2
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => [[[.,.],.],.] => 3
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => [[.,[.,.]],.] => 2
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => [[.,.],[.,.]] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,4,1,2] => [[.,[.,.]],[.,.]] => 2
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,1,2] => [[[.,[.,.]],.],.] => 3
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,3,4,1] => [[.,.],[.,[.,.]]] => 2
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => [[.,.],[[.,.],.]] => 2
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.] => 3
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [3,4,2,1] => [[[.,.],.],[.,.]] => 3
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,3,2,1] => [[[[.,.],.],.],.] => 4
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]] => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]] => 3
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => [[.,[.,[.,.]]],.] => 2
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => [[.,[.,.]],[.,.]] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]] => 2
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]] => 3
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]] => 2
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]] => 3
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.] => 4
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]] => 2
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.] => 3
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]] => 2
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => 2
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.] => 3
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]] => 2
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]] => 2
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => 3
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]] => 3
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]] => 4
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]] => 3
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]] => 4
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => 5
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [4,5,1,3,2] => [[.,[[.,.],.]],[.,.]] => 2
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.] => 3
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]] => 3
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]] => 3
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]] => 3
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.] => 4
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,2,1,4] => [[[.,.],.],[[.,.],.]] => 3
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => 4
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => 3
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]] => 2
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [5,4,6,1,2,3] => [[[.,[.,[.,.]]],.],[.,.]] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [4,6,5,1,2,3] => [[.,[.,[.,.]]],[[.,.],.]] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [5,6,4,1,2,3] => [[[.,[.,[.,.]]],.],[.,.]] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],.] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [3,4,5,6,1,2] => [[.,[.,.]],[.,[.,[.,.]]]] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [4,3,5,6,1,2] => [[[.,[.,.]],.],[.,[.,.]]] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [3,4,6,5,1,2] => [[.,[.,.]],[.,[[.,.],.]]] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [3,5,6,4,1,2] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [5,3,6,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [3,6,4,5,1,2] => [[.,[.,.]],[[.,[.,.]],.]] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [3,6,5,4,1,2] => [[.,[.,.]],[[[.,.],.],.]] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,3,4,1,2] => [[[[.,[.,.]],[.,.]],.],.] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [4,5,6,2,1,3] => [[[.,.],[.,.]],[.,[.,.]]] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [5,4,6,2,1,3] => [[[[.,.],[.,.]],.],[.,.]] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,6,5,2,1,3] => [[[.,.],[.,.]],[[.,.],.]] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [5,6,4,2,1,3] => [[[[.,.],[.,.]],.],[.,.]] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,5,4,2,1,3] => [[[[[.,.],[.,.]],.],.],.] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [3,5,6,1,2,4] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [5,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [3,4,5,1,6,2] => [[.,[.,.]],[.,[.,[.,.]]]] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [3,4,6,1,5,2] => [[.,[.,.]],[.,[[.,.],.]]] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [4,3,6,1,5,2] => [[[.,[.,.]],.],[[.,.],.]] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [3,5,4,1,6,2] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [3,6,4,1,5,2] => [[.,[.,.]],[[.,[.,.]],.]] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [4,6,3,1,5,2] => [[[.,[.,.]],.],[[.,.],.]] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [6,4,3,1,5,2] => [[[[.,[.,.]],.],[.,.]],.] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [5,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,.]] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,5,1,2,3,4] => [[[.,[.,[.,[.,.]]]],.],.] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [4,5,1,2,6,3] => [[.,[.,[.,.]]],[.,[.,.]]] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [4,6,1,2,5,3] => [[.,[.,[.,.]]],[[.,.],.]] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [6,4,1,2,5,3] => [[[.,[.,[.,.]]],[.,.]],.] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,4,1,5,6,2] => [[.,[.,.]],[.,[.,[.,.]]]] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,4,1,6,5,2] => [[.,[.,.]],[.,[[.,.],.]]] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [3,6,1,4,5,2] => [[.,[.,.]],[[.,[.,.]],.]] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [6,3,1,4,5,2] => [[[.,[.,.]],[.,[.,.]]],.] => 3
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]] => 2
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Description
The number of nodes on the left branch of a binary tree.
Also corresponds to ST000011 after applying the Tamari bijection between binary trees and Dyck path.
Also corresponds to ST000011 after applying the Tamari bijection between binary trees and Dyck path.
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
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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