Identifier
-
Mp00090:
Permutations
—cycle-as-one-line notation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001960: Permutations ⟶ ℤ
Values
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [2,3,1] => [2,1,3] => 0
[3,2,1] => [1,3,2] => [2,3,1] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 0
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [2,1,3,4] => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 0
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [2,1,3,4] => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 0
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 0
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 0
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 0
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 0
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [4,3,1,2,5] => 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => [4,3,1,2,5] => 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[1,5,4,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,1,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1,2,3,5] => 0
[2,4,1,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[2,4,1,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[2,4,5,1,3] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[2,4,5,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [4,3,1,2,5] => 1
[2,5,1,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[2,5,3,1,4] => [1,2,5,4,3] => [3,4,5,2,1] => [4,3,1,2,5] => 1
[2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[2,5,4,1,3] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0
[3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,1,2,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,1,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [2,5,1,3,4] => 0
[3,1,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [2,1,3,4,5] => 0
[3,1,5,2,4] => [1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => 1
[3,1,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 0
[3,2,1,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,2,1,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,2,4,1,5] => [1,3,4,2,5] => [5,2,4,3,1] => [2,5,1,3,4] => 0
[3,2,4,5,1] => [1,3,4,5,2] => [2,5,4,3,1] => [2,1,3,4,5] => 0
[3,2,5,1,4] => [1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => 1
[3,2,5,4,1] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 0
[3,4,1,2,5] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,4,1,5,2] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,4,2,1,5] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,4,2,5,1] => [1,3,2,4,5] => [5,4,2,3,1] => [2,4,5,1,3] => 0
[3,4,5,1,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 0
[3,4,5,2,1] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 0
[3,5,1,2,4] => [1,3,2,5,4] => [4,5,2,3,1] => [4,2,5,1,3] => 1
[3,5,1,4,2] => [1,3,2,5,4] => [4,5,2,3,1] => [4,2,5,1,3] => 1
[3,5,2,1,4] => [1,3,2,5,4] => [4,5,2,3,1] => [4,2,5,1,3] => 1
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Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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].
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].
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