- FindStat

Identifier

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2229 entries. <<<

Map

to binary tree: left brother = left child

Description

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

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.