- FindStat

Identifier

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤ (values match St000166The depth minus 1 of an ordered tree.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => [[.,.],[.,[.,[.,.]]]] => [[[]],[],[],[]] => 3

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

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

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

[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] => [[.,[.,.]],[.,[.,.]]] => [[[],[]],[],[]] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

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$F_{1} = q^{2}$

$F_{2} = q^{2} + q^{3}$

$F_{3} = q^{2} + 4\ q^{3} + q^{4}$

$F_{4} = q^{2} + 14\ q^{3} + 8\ q^{4} + q^{5}$

$F_{5} = q^{2} + 51\ q^{3} + 54\ q^{4} + 13\ q^{5} + q^{6}$

$F_{6} = q^{2} + 202\ q^{3} + 365\ q^{4} + 132\ q^{5} + 19\ q^{6} + q^{7}$

Description

The depth of an ordered tree.

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 $y$ .