- FindStat

Identifier

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000166: Ordered trees ⟶ ℤ (values match St000094The depth of an ordered tree.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to St000013The height of a Dyck path..

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.

Map

to ordered tree: right child = right brother

Description

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

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