- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ 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] => [[.,[.,.]],[.,.]] => [[[]],[[]]] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.