- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000884: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => 0

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

[2,1] => [2,1] => [1,2] => 0

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

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

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

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

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

[3,2,1] => [3,2,1] => [1,2,3] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].

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].