Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000083: Binary trees ⟶ ℤ
Values
[1,0,1,0] => [2,1] => [1,2] => [.,[.,.]] => 1
[1,1,0,0] => [1,2] => [1,2] => [.,[.,.]] => 1
[1,0,1,0,1,0] => [3,2,1] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,0,1,1,0,0] => [2,3,1] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,1,0,0,1,0] => [3,1,2] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,1,0,1,0,0] => [2,1,3] => [1,3,2] => [.,[[.,.],.]] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 2
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]] => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]] => 2
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [1,3,4,2] => [.,[[.,.],[.,.]]] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 3
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]] => 3
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]] => 3
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]] => 3
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]] => 3
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]] => 3
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]] => 3
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]] => 3
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 3
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]] => 3
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]] => 3
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [1,2,6,3,5,4] => [.,[.,[[.,[[.,.],.]],.]]] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
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Description
The number of left oriented leafs of a binary tree except the first one.
In other other words, this is the sum of canopee vector of the tree.
The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf.
This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.
In other other words, this is the sum of canopee vector of the tree.
The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf.
This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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 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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