- FindStat

Identifier

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[],[],[[],[[]]]] => [.,[.,[[.,[[.,.],.]],.]]] => [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] => 4

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

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

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

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

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

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

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

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

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

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

[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [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] => 4

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

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

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

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

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

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

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

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

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

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Description

The length of the longest increasing subsequence of the permutation.

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.