- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001584: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The area statistic between a Dyck path and its bounce path.
The bounce path Mp00099bounce path is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

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

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.