- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000058: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).