Identifier
-
Mp00132:
Dyck paths
—switch returns and last double rise⟶
Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤ
Values
[1,0] => [1,0] => [1] => [.,.] => 0
[1,0,1,0] => [1,0,1,0] => [2,1] => [[.,.],.] => 1
[1,1,0,0] => [1,1,0,0] => [1,2] => [.,[.,.]] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [2,1,3] => [[.,.],[.,.]] => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,3,2] => [.,[[.,.],.]] => 0
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [3,1,2] => [[.,.],[.,.]] => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [2,3,1] => [[.,[.,.]],.] => 2
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,2,3] => [.,[.,[.,.]]] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [[.,.],[[.,.],.]] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 0
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => [[.,.],[.,[.,.]]] => 1
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [[.,[.,.]],[.,.]] => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 0
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [[.,.],[[.,.],.]] => 1
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 0
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [[.,.],[.,[.,.]]] => 1
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [[.,[.,.]],[.,.]] => 2
[1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => [1,3,4,2] => [.,[[.,[.,.]],.]] => 0
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [[.,.],[.,[.,.]]] => 1
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [[.,[.,.]],[.,.]] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [[.,[.,[.,.]]],.] => 3
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 0
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 0
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 0
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => 3
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4,6] => [[.,.],[[.,.],[.,[.,.]]]] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => [.,[[.,.],[.,[[.,.],.]]]] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,4,2,6,5] => [.,[[.,[.,.]],[[.,.],.]]] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,1,2,3,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [[.,[.,.]],[.,[[.,.],.]]] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [[.,[.,[.,.]]],[[.,.],.]] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,5,2,6,4] => [[.,.],[[.,.],[[.,.],.]]] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,2,6,3,5] => [[.,.],[.,[[.,.],[.,.]]]] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [[.,.],[[.,.],[.,[.,.]]]] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [[.,.],[[.,[.,.]],[.,.]]] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,5,2,6,4] => [.,[[.,[.,.]],[[.,.],.]]] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,1,2,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [[.,[.,.]],[[.,[.,.]],.]] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [[.,[.,[.,.]]],[[.,.],.]] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,6,2,3,5] => [[.,.],[[.,.],[.,[.,.]]]] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => [.,[[.,.],[.,[.,[.,.]]]]] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [[.,.],[.,[[.,.],[.,.]]]] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [[.,[.,.]],[[.,.],[.,.]]] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [[.,[.,.]],[[.,[.,.]],.]] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [[.,[.,[.,.]]],[[.,.],.]] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]] => 0
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Description
The size of the left subtree of a binary tree.
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
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.
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.
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.
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