Identifier
-
Mp00016:
Binary trees
—left-right symmetry⟶
Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000354: Permutations ⟶ ℤ
Values
[.,[.,.]] => [[.,.],.] => [.,[.,.]] => [2,1] => 1
[[.,.],.] => [.,[.,.]] => [[.,.],.] => [1,2] => 0
[.,[.,[.,.]]] => [[[.,.],.],.] => [.,[[.,.],.]] => [2,3,1] => 1
[.,[[.,.],.]] => [[.,[.,.]],.] => [.,[.,[.,.]]] => [3,2,1] => 2
[[.,.],[.,.]] => [[.,.],[.,.]] => [[[.,.],.],.] => [1,2,3] => 0
[[.,[.,.]],.] => [.,[[.,.],.]] => [[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [.,[.,[.,.]]] => [[.,.],[.,.]] => [1,3,2] => 1
[.,[.,[.,[.,.]]]] => [[[[.,.],.],.],.] => [.,[[[.,.],.],.]] => [2,3,4,1] => 1
[.,[.,[[.,.],.]]] => [[[.,[.,.]],.],.] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 2
[.,[[.,.],[.,.]]] => [[[.,.],[.,.]],.] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 2
[.,[[.,[.,.]],.]] => [[.,[[.,.],.]],.] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 2
[.,[[[.,.],.],.]] => [[.,[.,[.,.]]],.] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 3
[[.,.],[.,[.,.]]] => [[[.,.],.],[.,.]] => [[[[.,.],.],.],.] => [1,2,3,4] => 0
[[.,.],[[.,.],.]] => [[.,[.,.]],[.,.]] => [[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[.,[.,.]],[.,.]] => [[.,.],[[.,.],.]] => [[[.,.],[.,.]],.] => [1,3,2,4] => 1
[[[.,.],.],[.,.]] => [[.,.],[.,[.,.]]] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.] => [.,[[[.,.],.],.]] => [[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[.,[[.,.],.]],.] => [.,[[.,[.,.]],.]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 2
[[[.,.],[.,.]],.] => [.,[[.,.],[.,.]]] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 2
[[[.,[.,.]],.],.] => [.,[.,[[.,.],.]]] => [[.,.],[[.,.],.]] => [1,3,4,2] => 1
[[[[.,.],.],.],.] => [.,[.,[.,[.,.]]]] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 2
[.,[.,[.,[.,[.,.]]]]] => [[[[[.,.],.],.],.],.] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[.,[.,[.,[[.,.],.]]]] => [[[[.,[.,.]],.],.],.] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 2
[.,[.,[[.,.],[.,.]]]] => [[[[.,.],[.,.]],.],.] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 2
[.,[.,[[.,[.,.]],.]]] => [[[.,[[.,.],.]],.],.] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 2
[.,[.,[[[.,.],.],.]]] => [[[.,[.,[.,.]]],.],.] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 3
[.,[[.,.],[.,[.,.]]]] => [[[[.,.],.],[.,.]],.] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 2
[.,[[.,.],[[.,.],.]]] => [[[.,[.,.]],[.,.]],.] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 3
[.,[[.,[.,.]],[.,.]]] => [[[.,.],[[.,.],.]],.] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 2
[.,[[[.,.],.],[.,.]]] => [[[.,.],[.,[.,.]]],.] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 3
[.,[[.,[.,[.,.]]],.]] => [[.,[[[.,.],.],.]],.] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 2
[.,[[.,[[.,.],.]],.]] => [[.,[[.,[.,.]],.]],.] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 3
[.,[[[.,.],[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 3
[.,[[[.,[.,.]],.],.]] => [[.,[.,[[.,.],.]]],.] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 3
[.,[[[[.,.],.],.],.]] => [[.,[.,[.,[.,.]]]],.] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 4
[[.,.],[.,[.,[.,.]]]] => [[[[.,.],.],.],[.,.]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
[[.,.],[.,[[.,.],.]]] => [[[.,[.,.]],.],[.,.]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[.,.],[[.,.],[.,.]]] => [[[.,.],[.,.]],[.,.]] => [[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 1
[[.,.],[[.,[.,.]],.]] => [[.,[[.,.],.]],[.,.]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 1
[[.,.],[[[.,.],.],.]] => [[.,[.,[.,.]]],[.,.]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2
[[.,[.,.]],[.,[.,.]]] => [[[.,.],.],[[.,.],.]] => [[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 1
[[.,[.,.]],[[.,.],.]] => [[.,[.,.]],[[.,.],.]] => [[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 2
[[[.,.],.],[.,[.,.]]] => [[[.,.],.],[.,[.,.]]] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1
[[[.,.],.],[[.,.],.]] => [[.,[.,.]],[.,[.,.]]] => [[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 2
[[.,[.,[.,.]]],[.,.]] => [[.,.],[[[.,.],.],.]] => [[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 1
[[.,[[.,.],.]],[.,.]] => [[.,.],[[.,[.,.]],.]] => [[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 2
[[[.,.],[.,.]],[.,.]] => [[.,.],[[.,.],[.,.]]] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]] => [[.,.],[.,[[.,.],.]]] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 1
[[[[.,.],.],.],[.,.]] => [[.,.],[.,[.,[.,.]]]] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2
[[.,[.,[.,[.,.]]]],.] => [.,[[[[.,.],.],.],.]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [.,[[[.,[.,.]],.],.]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 2
[[.,[[.,.],[.,.]]],.] => [.,[[[.,.],[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 2
[[.,[[.,[.,.]],.]],.] => [.,[[.,[[.,.],.]],.]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[.,[[[.,.],.],.]],.] => [.,[[.,[.,[.,.]]],.]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 3
[[[.,.],[.,[.,.]]],.] => [.,[[[.,.],.],[.,.]]] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 2
[[[.,.],[[.,.],.]],.] => [.,[[.,[.,.]],[.,.]]] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 3
[[[.,[.,.]],[.,.]],.] => [.,[[.,.],[[.,.],.]]] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 2
[[[[.,.],.],[.,.]],.] => [.,[[.,.],[.,[.,.]]]] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 3
[[[.,[.,[.,.]]],.],.] => [.,[.,[[[.,.],.],.]]] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1
[[[.,[[.,.],.]],.],.] => [.,[.,[[.,[.,.]],.]]] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 2
[[[[.,.],[.,.]],.],.] => [.,[.,[[.,.],[.,.]]]] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 2
[[[[.,[.,.]],.],.],.] => [.,[.,[.,[[.,.],.]]]] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 2
[[[[[.,.],.],.],.],.] => [.,[.,[.,[.,[.,.]]]]] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 3
[.,[.,[.,[.,[.,[.,.]]]]]] => [[[[[[.,.],.],.],.],.],.] => [.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [[[[[.,[.,.]],.],.],.],.] => [.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [[[[[.,.],[.,.]],.],.],.] => [.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [[[[.,[[.,.],.]],.],.],.] => [.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [[[[.,[.,[.,.]]],.],.],.] => [.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [[[[[.,.],.],[.,.]],.],.] => [.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [[[[.,[.,.]],[.,.]],.],.] => [.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [[[[.,.],[[.,.],.]],.],.] => [.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 2
[.,[.,[[[.,.],.],[.,.]]]] => [[[[.,.],[.,[.,.]]],.],.] => [.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [[[.,[[[.,.],.],.]],.],.] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [[[.,[[.,[.,.]],.]],.],.] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [[[.,[[.,.],[.,.]]],.],.] => [.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [[[.,[.,[[.,.],.]]],.],.] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 3
[.,[.,[[[[.,.],.],.],.]]] => [[[.,[.,[.,[.,.]]]],.],.] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 4
[.,[[.,.],[.,[.,[.,.]]]]] => [[[[[.,.],.],.],[.,.]],.] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2
[.,[[.,.],[.,[[.,.],.]]]] => [[[[.,[.,.]],.],[.,.]],.] => [.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 3
[.,[[.,.],[[.,.],[.,.]]]] => [[[[.,.],[.,.]],[.,.]],.] => [.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 3
[.,[[.,.],[[.,[.,.]],.]]] => [[[.,[[.,.],.]],[.,.]],.] => [.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 3
[.,[[.,.],[[[.,.],.],.]]] => [[[.,[.,[.,.]]],[.,.]],.] => [.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 4
[.,[[.,[.,.]],[.,[.,.]]]] => [[[[.,.],.],[[.,.],.]],.] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 2
[.,[[.,[.,.]],[[.,.],.]]] => [[[.,[.,.]],[[.,.],.]],.] => [.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 3
[.,[[[.,.],.],[.,[.,.]]]] => [[[[.,.],.],[.,[.,.]]],.] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 3
[.,[[[.,.],.],[[.,.],.]]] => [[[.,[.,.]],[.,[.,.]]],.] => [.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 4
[.,[[.,[.,[.,.]]],[.,.]]] => [[[.,.],[[[.,.],.],.]],.] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 2
[.,[[.,[[.,.],.]],[.,.]]] => [[[.,.],[[.,[.,.]],.]],.] => [.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [[[.,.],[[.,.],[.,.]]],.] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [[[.,.],[.,[[.,.],.]]],.] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 3
[.,[[[[.,.],.],.],[.,.]]] => [[[.,.],[.,[.,[.,.]]]],.] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 4
[.,[[.,[.,[.,[.,.]]]],.]] => [[.,[[[[.,.],.],.],.]],.] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 2
[.,[[.,[.,[[.,.],.]]],.]] => [[.,[[[.,[.,.]],.],.]],.] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [[.,[[[.,.],[.,.]],.]],.] => [.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [[.,[[.,[[.,.],.]],.]],.] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 3
[.,[[.,[[[.,.],.],.]],.]] => [[.,[[.,[.,[.,.]]],.]],.] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 4
[.,[[[.,.],[.,[.,.]]],.]] => [[.,[[[.,.],.],[.,.]]],.] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 3
[.,[[[.,.],[[.,.],.]],.]] => [[.,[[.,[.,.]],[.,.]]],.] => [.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 4
[.,[[[.,[.,.]],[.,.]],.]] => [[.,[[.,.],[[.,.],.]]],.] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 3
[.,[[[[.,.],.],[.,.]],.]] => [[.,[[.,.],[.,[.,.]]]],.] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 4
[.,[[[.,[.,[.,.]]],.],.]] => [[.,[.,[[[.,.],.],.]]],.] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 3
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of recoils of a permutation.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
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
left-right symmetry
Description
Return the left-right symmetrized tree of a binary tree.
Map
left rotate
Description
Return the result of left rotation applied to a binary tree.
Left rotation on binary trees is defined as follows: Let $T$ be a binary tree such that the right child of the root of $T$ is a node. Let $A$ be the left child of the root of $T$, and let $B$ and $C$ be the left and right children of the right child of the root of $T$. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the left rotation of $T$ is the binary tree in which the right child of the root is $C$, whereas the left child of the root is a node whose left and right children are $A$ and $B$.
Left rotation on binary trees is defined as follows: Let $T$ be a binary tree such that the right child of the root of $T$ is a node. Let $A$ be the left child of the root of $T$, and let $B$ and $C$ be the left and right children of the right child of the root of $T$. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the left rotation of $T$ is the binary tree in which the right child of the root is $C$, whereas the left child of the root is a node whose left and right children are $A$ and $B$.
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