- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ 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] => [3,4,1,2] => [[.,[.,.]],[.,.]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 10988 entries. <<<

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.

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.