- FindStat

Identifier

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤ

Values

[[]] => [.,.] => [1] => 1

[[],[]] => [.,[.,.]] => [2,1] => 2

[[[]]] => [[.,.],.] => [1,2] => 1

[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => 3

[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => 2

[[[]],[]] => [[.,.],[.,.]] => [3,1,2] => 2

[[[],[]]] => [[.,[.,.]],.] => [2,1,3] => 1

[[[[]]]] => [[[.,.],.],.] => [1,2,3] => 1

[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 4

[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 3

[[],[[]],[]] => [.,[[.,.],[.,.]]] => [4,2,3,1] => 3

[[],[[],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 2

[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => 2

[[[]],[],[]] => [[.,.],[.,[.,.]]] => [4,3,1,2] => 3

[[[]],[[]]] => [[.,.],[[.,.],.]] => [3,4,1,2] => 2

[[[],[]],[]] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 2

[[[[]]],[]] => [[[.,.],.],[.,.]] => [4,1,2,3] => 2

[[[],[],[]]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 1

[[[],[[]]]] => [[.,[[.,.],.]],.] => [2,3,1,4] => 1

[[[[]],[]]] => [[[.,.],[.,.]],.] => [3,1,2,4] => 1

[[[[],[]]]] => [[[.,[.,.]],.],.] => [2,1,3,4] => 1

[[[[[]]]]] => [[[[.,.],.],.],.] => [1,2,3,4] => 1

[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 5

[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 4

[[],[],[[]],[]] => [.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 4

[[],[],[[],[]]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 3

[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 3

[[],[[]],[],[]] => [.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 4

[[],[[]],[[]]] => [.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 3

[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 3

[[],[[[]]],[]] => [.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 3

[[],[[],[],[]]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 2

[[],[[],[[]]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 2

[[],[[[]],[]]] => [.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 2

[[],[[[],[]]]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 2

[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 2

[[[]],[],[],[]] => [[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 4

[[[]],[],[[]]] => [[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 3

[[[]],[[]],[]] => [[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 3

[[[]],[[],[]]] => [[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 2

[[[]],[[[]]]] => [[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 2

[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 3

[[[[]]],[],[]] => [[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 3

[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 2

[[[[]]],[[]]] => [[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 2

[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 2

[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 2

[[[[]],[]],[]] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 2

[[[[],[]]],[]] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 2

[[[[[]]]],[]] => [[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 2

[[[],[],[],[]]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1

[[[],[],[[]]]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1

[[[],[[]],[]]] => [[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 1

[[[],[[],[]]]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1

[[[],[[[]]]]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1

[[[[]],[],[]]] => [[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 1

[[[[]],[[]]]] => [[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 1

[[[[],[]],[]]] => [[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 1

[[[[[]]],[]]] => [[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 1

[[[[],[],[]]]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 1

[[[[],[[]]]]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 1

[[[[[]],[]]]] => [[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 1

[[[[[],[]]]]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1

[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1

[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 6

[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 5

[[],[],[],[[]],[]] => [.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 5

[[],[],[],[[],[]]] => [.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 4

[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 4

[[],[],[[]],[],[]] => [.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 5

[[],[],[[]],[[]]] => [.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 4

[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 4

[[],[],[[[]]],[]] => [.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 4

[[],[],[[],[],[]]] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 3

[[],[],[[],[[]]]] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 3

[[],[],[[[]],[]]] => [.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 3

[[],[],[[[],[]]]] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 3

[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 3

[[],[[]],[],[],[]] => [.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 5

[[],[[]],[],[[]]] => [.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 4

[[],[[]],[[]],[]] => [.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 4

[[],[[]],[[],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 3

[[],[[]],[[[]]]] => [.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 3

[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 4

[[],[[[]]],[],[]] => [.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 4

[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 3

[[],[[[]]],[[]]] => [.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 3

[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 3

[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 3

[[],[[[]],[]],[]] => [.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 3

[[],[[[],[]]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 3

[[],[[[[]]]],[]] => [.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 3

[[],[[],[],[],[]]] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 2

[[],[[],[],[[]]]] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 2

[[],[[],[[]],[]]] => [.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 2

[[],[[],[[],[]]]] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 2

[[],[[],[[[]]]]] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 2

[[],[[[]],[],[]]] => [.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 2

[[],[[[]],[[]]]] => [.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 2

[[],[[[],[]],[]]] => [.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 2

[[],[[[[]]],[]]] => [.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 2

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Description

The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].

Map

to binary tree: right brother = right child

Description

Return a binary tree of size $n-1$ (where $n$ is the size of an ordered tree $t$) obtained from $t$ by the following recursive rule:
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left child of $y$,
and removing the root of $t$.

Map

to 132-avoiding permutation

Description

Return a 132-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 maximal element of the Sylvester class.