- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 10755 entries. <<<

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

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

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].