- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤ (values match St000196The number of occurrences of the contiguous pattern [[.,.],[.,.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7203 entries. <<<

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Description

The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

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.