- FindStat

Identifier

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)

Values

[[]] => [.,.] => [1] => 0

[[],[]] => [[.,.],.] => [1,2] => 0

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

[[],[],[]] => [[[.,.],.],.] => [1,2,3] => 0

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

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

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

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

[[],[],[],[]] => [[[[.,.],.],.],.] => [1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[[],[],[],[],[]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[],[],[],[],[],[]] => [[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

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 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.