- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 18248 entries. <<<

Map

to car permutation

Description

Maps a parking function to the permutation of cars that correspond to that parking function.
For example, the image

$[2, 4, 5, 6, 3, 1, 7]$ means that car 2 takes spots 1, car 4 takes spot 2, and so on.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

descent views to invisible inversion bottoms

Description

Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
An invisible inversion of a permutation

$\sigma$ is a pair

$i < j$ such that

$i < \sigma(j) < \sigma(i)$ . The element

$\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation

$\pi$ is an element

$\pi(j)$ such that

$\pi(i+1) < \pi(j) < \pi(i)$ , and additionally the smallest element in the decreasing run containing

$\pi(i)$ is smaller than the smallest element in the decreasing run containing

$\pi(j)$ .
This map is a bijection

$\chi:\mathfrak S_n \to \mathfrak S_n$ , such that

the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$ ,
the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$ ,
the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$ ,
the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$ , and
the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$ .