- FindStat

Identifier

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

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

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 $y$.

Map

to labelling permutation

Description

Returns the labelling of the support Dyck path of the parking function.