- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5913 entries. <<<

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

Tanimoto

Description

Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.

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.