- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ 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] => [1,2,3] => [.,[.,[.,.]]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 6996 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

inverse toric promotion

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.

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.