- FindStat

Identifier

Mp00069: Permutations —complement⟶ 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] => [2,1] => [[.,.],.] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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.