Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤ (values match St000092The number of outer peaks of a permutation.)
Values
[1,0,1,0] => [2,1] => [[.,.],.] => [1,2] => 0
[1,1,0,0] => [1,2] => [.,[.,.]] => [2,1] => 0
[1,0,1,0,1,0] => [2,1,3] => [[.,.],[.,.]] => [1,3,2] => 0
[1,0,1,1,0,0] => [2,3,1] => [[.,[.,.]],.] => [2,1,3] => 1
[1,1,0,0,1,0] => [3,1,2] => [[.,.],[.,.]] => [1,3,2] => 0
[1,1,0,1,0,0] => [1,3,2] => [.,[[.,.],.]] => [2,3,1] => 0
[1,1,1,0,0,0] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [[.,.],[[.,.],.]] => [1,3,4,2] => 0
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 0
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [[.,.],[[.,.],.]] => [1,3,4,2] => 0
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 0
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 0
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 0
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 0
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 0
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Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is St000099The number of valleys of a permutation, including the boundary..
The number of valleys including the boundary is St000099The number of valleys of a permutation, including the boundary..
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.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
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.
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