- FindStat

Identifier

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

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2229 entries. <<<

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.