- FindStat

Identifier

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 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 signed permutations.
This map is a surjection onto the set of involutions, i.e., the set of signed permutations $\pi$ for which $\pi = \pi^{-1}$.

Map

permutation

Description

The permutation obtained by forgetting the colours.

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.