- FindStat

Identifier

Mp00066: Permutations —inverse⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 33360 entries. <<<

Map

inverse

Description

Sends a permutation to its inverse.

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.