- FindStat

Identifier

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,.]]]]] => [[[[[[]]]]]] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [[[[[[[]]]]]]] => 5

[.,[.,[.,[.,[[.,.],.]]]]] => [[[[[[],[]]]]]] => 4

[.,[.,[.,[[.,.],[.,.]]]]] => [[[[[],[[]]]]]] => 4

[.,[.,[.,[[.,[.,.]],.]]]] => [[[[[[]],[]]]]] => 4

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

[.,[.,[[.,.],[.,[.,.]]]]] => [[[[],[[[]]]]]] => 4

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

[.,[.,[[.,[.,.]],[.,.]]]] => [[[[[]],[[]]]]] => 4

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

[.,[.,[[.,[.,[.,.]]],.]]] => [[[[[[]]],[]]]] => 4

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

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

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

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

[.,[[.,.],[.,[.,[.,.]]]]] => [[[],[[[[]]]]]] => 4

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

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

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

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

[.,[[.,[.,.]],[.,[.,.]]]] => [[[[]],[[[]]]]] => 4

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

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

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

[.,[[.,[.,[.,.]]],[.,.]]] => [[[[[]]],[[]]]] => 4

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

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

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

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

[.,[[.,[.,[.,[.,.]]]],.]] => [[[[[[]]]],[]]] => 4

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{2}$

$F_{4} = 1 + 6\ q + 6\ q^{2} + q^{3}$

$F_{5} = 1 + 10\ q + 20\ q^{2} + 10\ q^{3} + q^{4}$

$F_{6} = 1 + 15\ q + 50\ q^{2} + 50\ q^{3} + 15\ q^{4} + q^{5}$

Description

The number of internal nodes of an ordered tree.
A node is internal if it is neither the root nor a leaf.

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 $y$ .