Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000083: Binary trees ⟶ ℤ
Values
[1,0,1,0] => [1,2] => [1,2] => [.,[.,.]] => 1
[1,1,0,0] => [2,1] => [1,2] => [.,[.,.]] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,0,1,1,0,0] => [1,3,2] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,1,0,0,1,0] => [2,1,3] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,1,0,1,0,0] => [2,3,1] => [1,2,3] => [.,[.,[.,.]]] => 2
[1,1,1,0,0,0] => [3,2,1] => [1,3,2] => [.,[[.,.],.]] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 3
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 2
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]] => 2
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 2
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 3
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 3
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 3
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]] => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 4
>>> Load all 195 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to 312-avoiding permutation
Description
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!