- FindStat

Identifier

Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5913 entries. <<<

Map

weak order rowmotion

Description

Return the reversal of the permutation obtained by inverting the corresponding Laguerre heap.
This map is the composite of Mp00241invert Laguerre heap and Mp00064reverse.
Conjecturally, it is also the rowmotion on the weak order:
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$.
Note that the dynamics of weak order rowmotion is poorly understood. A collection of nontrivial homomesies is described in Corollary 6.14 of [4].

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.