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Identifier

Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions

Images

[1] => [1] => {{1}} => {{1}}

[1,2] => [2,1] => {{1,2}} => {{1,2}}

[2,1] => [1,2] => {{1},{2}} => {{1},{2}}

[1,2,3] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[1,3,2] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[2,1,3] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[2,3,1] => [2,1,3] => {{1,2},{3}} => {{1},{2,3}}

[3,1,2] => [1,3,2] => {{1},{2,3}} => {{1,2},{3}}

[3,2,1] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

[1,2,3,4] => [4,3,2,1] => {{1,4},{2,3}} => {{1,4},{2,3}}

[1,2,4,3] => [3,4,2,1] => {{1,2,3,4}} => {{1,2,3,4}}

[1,3,2,4] => [4,2,3,1] => {{1,4},{2},{3}} => {{1,4},{2},{3}}

[1,3,4,2] => [3,2,4,1] => {{1,3,4},{2}} => {{1,2,4},{3}}

[1,4,2,3] => [2,4,3,1] => {{1,2,4},{3}} => {{1,3,4},{2}}

[1,4,3,2] => [2,3,4,1] => {{1,2,3,4}} => {{1,2,3,4}}

[2,1,3,4] => [4,3,1,2] => {{1,2,3,4}} => {{1,2,3,4}}

[2,1,4,3] => [3,4,1,2] => {{1,3},{2,4}} => {{1,3},{2,4}}

[2,3,1,4] => [4,2,1,3] => {{1,3,4},{2}} => {{1,2,4},{3}}

[2,3,4,1] => [3,2,1,4] => {{1,3},{2},{4}} => {{1},{2,4},{3}}

[2,4,1,3] => [2,1,4,3] => {{1,2},{3,4}} => {{1,2},{3,4}}

[2,4,3,1] => [2,1,3,4] => {{1,2},{3},{4}} => {{1},{2},{3,4}}

[3,1,2,4] => [4,1,3,2] => {{1,2,4},{3}} => {{1,3,4},{2}}

[3,1,4,2] => [1,3,2,4] => {{1},{2,3},{4}} => {{1},{2,3},{4}}

[3,2,1,4] => [4,1,2,3] => {{1,2,3,4}} => {{1,2,3,4}}

[3,2,4,1] => [2,3,1,4] => {{1,2,3},{4}} => {{1},{2,3,4}}

[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}} => {{1,2,3,4}}

[3,4,2,1] => [3,1,2,4] => {{1,2,3},{4}} => {{1},{2,3,4}}

[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}} => {{1,3},{2},{4}}

[4,1,3,2] => [1,4,2,3] => {{1},{2,3,4}} => {{1,2,3},{4}}

[4,2,1,3] => [1,2,4,3] => {{1},{2},{3,4}} => {{1,2},{3},{4}}

[4,2,3,1] => [2,4,1,3] => {{1,2,3,4}} => {{1,2,3,4}}

[4,3,1,2] => [1,3,4,2] => {{1},{2,3,4}} => {{1,2,3},{4}}

[4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}} => {{1},{2},{3},{4}}

[1,2,3,4,5] => [5,4,3,2,1] => {{1,5},{2,4},{3}} => {{1,5},{2,4},{3}}

[1,2,3,5,4] => [4,5,3,2,1] => {{1,2,4,5},{3}} => {{1,2,4,5},{3}}

[1,2,4,3,5] => [5,3,4,2,1] => {{1,5},{2,3,4}} => {{1,5},{2,3,4}}

[1,2,4,5,3] => [4,3,5,2,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[1,2,5,3,4] => [3,5,4,2,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[1,2,5,4,3] => [3,4,5,2,1] => {{1,3,5},{2,4}} => {{1,3,5},{2,4}}

[1,3,2,4,5] => [5,4,2,3,1] => {{1,5},{2,3,4}} => {{1,5},{2,3,4}}

[1,3,2,5,4] => [4,5,2,3,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[1,3,4,2,5] => [5,3,2,4,1] => {{1,5},{2,3},{4}} => {{1,5},{2},{3,4}}

[1,3,4,5,2] => [4,3,2,5,1] => {{1,4,5},{2,3}} => {{1,2,5},{3,4}}

[1,3,5,2,4] => [3,2,5,4,1] => {{1,3,5},{2},{4}} => {{1,3,5},{2},{4}}

[1,3,5,4,2] => [3,2,4,5,1] => {{1,3,4,5},{2}} => {{1,2,3,5},{4}}

[1,4,2,3,5] => [5,2,4,3,1] => {{1,5},{2},{3,4}} => {{1,5},{2,3},{4}}

[1,4,2,5,3] => [2,4,3,5,1] => {{1,2,4,5},{3}} => {{1,2,4,5},{3}}

[1,4,3,2,5] => [5,2,3,4,1] => {{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}}

[1,4,3,5,2] => [3,4,2,5,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[1,4,5,2,3] => [4,2,5,3,1] => {{1,3,4,5},{2}} => {{1,2,3,5},{4}}

[1,4,5,3,2] => [4,2,3,5,1] => {{1,4,5},{2},{3}} => {{1,2,5},{3},{4}}

[1,5,2,3,4] => [2,5,4,3,1] => {{1,2,5},{3,4}} => {{1,4,5},{2,3}}

[1,5,2,4,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}} => {{1,4,5},{2},{3}}

[1,5,3,2,4] => [2,3,5,4,1] => {{1,2,3,5},{4}} => {{1,3,4,5},{2}}

[1,5,3,4,2] => [3,5,2,4,1] => {{1,2,3,5},{4}} => {{1,3,4,5},{2}}

[1,5,4,2,3] => [2,4,5,3,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[1,5,4,3,2] => [2,3,4,5,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[2,1,3,4,5] => [5,4,3,1,2] => {{1,2,4,5},{3}} => {{1,2,4,5},{3}}

[2,1,3,5,4] => [4,5,3,1,2] => {{1,4},{2,5},{3}} => {{1,4},{2,5},{3}}

[2,1,4,3,5] => [5,3,4,1,2] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[2,1,4,5,3] => [4,3,5,1,2] => {{1,4},{2,3,5}} => {{1,3,4},{2,5}}

[2,1,5,3,4] => [3,5,4,1,2] => {{1,3,4},{2,5}} => {{1,4},{2,3,5}}

[2,1,5,4,3] => [3,4,5,1,2] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[2,3,1,4,5] => [5,4,2,1,3] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[2,3,1,5,4] => [4,5,2,1,3] => {{1,4},{2,3,5}} => {{1,3,4},{2,5}}

[2,3,4,1,5] => [5,3,2,1,4] => {{1,4,5},{2,3}} => {{1,2,5},{3,4}}

[2,3,4,5,1] => [4,3,2,1,5] => {{1,4},{2,3},{5}} => {{1},{2,5},{3,4}}

[2,3,5,1,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}} => {{1,2},{3,5},{4}}

[2,3,5,4,1] => [3,2,1,4,5] => {{1,3},{2},{4},{5}} => {{1},{2},{3,5},{4}}

[2,4,1,3,5] => [5,2,1,4,3] => {{1,3,5},{2},{4}} => {{1,3,5},{2},{4}}

[2,4,1,5,3] => [2,1,4,3,5] => {{1,2},{3,4},{5}} => {{1},{2,3},{4,5}}

[2,4,3,1,5] => [5,2,1,3,4] => {{1,3,4,5},{2}} => {{1,2,3,5},{4}}

[2,4,3,5,1] => [3,4,2,1,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}}

[2,4,5,1,3] => [4,2,1,5,3] => {{1,3,4,5},{2}} => {{1,2,3,5},{4}}

[2,4,5,3,1] => [4,2,1,3,5] => {{1,3,4},{2},{5}} => {{1},{2,3,5},{4}}

[2,5,1,3,4] => [2,1,5,4,3] => {{1,2},{3,5},{4}} => {{1,3},{2},{4,5}}

[2,5,1,4,3] => [2,1,5,3,4] => {{1,2},{3,4,5}} => {{1,2,3},{4,5}}

[2,5,3,1,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}} => {{1,2},{3},{4,5}}

[2,5,3,4,1] => [3,5,2,1,4] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[2,5,4,1,3] => [2,1,4,5,3] => {{1,2},{3,4,5}} => {{1,2,3},{4,5}}

[2,5,4,3,1] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => {{1},{2},{3},{4,5}}

[3,1,2,4,5] => [5,4,1,3,2] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[3,1,2,5,4] => [4,5,1,3,2] => {{1,3,4},{2,5}} => {{1,4},{2,3,5}}

[3,1,4,2,5] => [5,1,3,2,4] => {{1,2,4,5},{3}} => {{1,2,4,5},{3}}

[3,1,4,5,2] => [4,1,3,2,5] => {{1,2,4},{3},{5}} => {{1},{2,4,5},{3}}

[3,1,5,2,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}} => {{1,2},{3,4},{5}}

[3,1,5,4,2] => [1,3,2,4,5] => {{1},{2,3},{4},{5}} => {{1},{2},{3,4},{5}}

[3,2,1,4,5] => [5,4,1,2,3] => {{1,3,5},{2,4}} => {{1,3,5},{2,4}}

[3,2,1,5,4] => [4,5,1,2,3] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[3,2,4,1,5] => [5,2,3,1,4] => {{1,4,5},{2},{3}} => {{1,2,5},{3},{4}}

[3,2,4,5,1] => [4,2,3,1,5] => {{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}}

[3,2,5,1,4] => [2,3,1,5,4] => {{1,2,3},{4,5}} => {{1,2},{3,4,5}}

[3,2,5,4,1] => [2,3,1,4,5] => {{1,2,3},{4},{5}} => {{1},{2},{3,4,5}}

[3,4,1,2,5] => [5,3,1,4,2] => {{1,2,3,5},{4}} => {{1,3,4,5},{2}}

[3,4,1,5,2] => [3,1,4,2,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}}

[3,4,2,1,5] => [5,3,1,2,4] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[3,4,2,5,1] => [3,2,4,1,5] => {{1,3,4},{2},{5}} => {{1},{2,3,5},{4}}

[3,4,5,1,2] => [4,3,1,5,2] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

[3,4,5,2,1] => [4,3,1,2,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}}

[3,5,1,2,4] => [3,1,5,4,2] => {{1,2,3,5},{4}} => {{1,3,4,5},{2}}

[3,5,1,4,2] => [3,1,5,2,4] => {{1,2,3,4,5}} => {{1,2,3,4,5}}

>>> 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 cycle type

Description

Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].

Map

complement

Description

The complement of a set partition obtained by replacing $i$ with $n+1-i$.