- FindStat

Identifier

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.

Map

to ordered tree: left child = left brother

Description

Return an ordered tree of size $n+1$ by the following recursive rule:

if $x$ is the left child of $y$, $x$ becomes the left brother of $y$,
if $x$ is the right child of $y$, $x$ becomes the last child of $y$.

Map

binary search tree: left to right

Description

Return the shape of the binary search tree of the permutation as a non labelled binary tree.