- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7229 entries. <<<

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation

$\sigma$ is inverse fireworks if its inverse avoids the vincular pattern

$3-12$ . The inverse fireworks map sends any permutation

$\sigma$ to an inverse fireworks permutation that is below

$\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

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.