- FindStat

Identifier

Mp00159: Permutations —Demazure product with 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] => [3,2,1] => [[[.,.],.],.]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 36768 entries. <<<

Map

Demazure product with inverse

Description

This map sends a permutation

$\pi$ to

$\pi^{-1} \star \pi$ where

$\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations

$\pi$ for which

$\pi = \pi^{-1}$ .

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.